bells are one of the oldest musical instruments which still play an important cultural role .they were invented in china but today their sound announces major events all around the world . depending on the region bellsare mounted in a number of different ways basing on local customs and tradition . in europewe can encounter three characteristic mounting layouts : central european , english and spanish .in central europe bells usually tilt on their axis with maximum amplitude of oscillations below degrees . in the english systemthe amplitude of oscillations is greater and bells perform nearly a complete rotations in both directions .conversely , in the spanish system bells rotate continuously in the same direction .all these mounting layouts were developed throughout centuries basing on experience and intuition of bell - founders and craftsmen .it is common that the bells are cast using casting mould passed down for ages from father to son and so forth .although the design of a bell , its yoke , clapper and a belfry has been being improved for ages , proper modeling of their dynamics is still a challenging task .the dynamics a yoke - bell - clapper system is extremely complex and difficult to analyze due to nonlinear characteristic , repetitive impacts and complicated excitation .moreover , depending on the mounting layout different dynamical states can be observed and each type of yoke has its own specific properties .however , already in 19th century wilhelm veltmann tried to describe mathematically the behavior of the famous emperors bell in the cologne cathedral .he used simplified equations of motion and explained why the clapper always remained on the middle axis of the bell instead of striking its shell .his model was developed basing on the equations of a double physical pendulum .heyman and threlfrall use similar model to estimate inertia forces induced by a swinging bell .the knowledge of loads induced by ringing bells is crucial during the design and restoration processes of belfries .that is why there is a number of works considering dynamic interactions between bell towers and bells mounted in different manners .muller , steiner and schutz focus on central european mounting system which is also considered in the german din standard .we can find similar studies concerning the spanish system and the english system .ivorra et .present the comparison between the three mounting layouts and prove that in the spanish system forces transmitted to the supporting structure are significantly lower than in the other two .results presented in show that in many cases we can improve bells working conditions slightly modifying the yoke or its support .apart from the studies concerning interactions between bells and their supports there is a number of publications focusing on the dynamics of the instrument itself or the clapper to the bell collisions .klemenc et .al . contributed with a series of papers devoted to the analysis of the clapper - to - bell impacts .authors investigate the consequences of the repetitive hits and compare experimental data with numerical results obtained from the finite - element model .presented results prove that full - scale finite - element model is able to reproduce the effect of collisions but requires long computational times and complex , detailed models .therefore , it would be difficult to use such models to analyze the dynamics of bells .because of that , recently we observe the tendency to use hybrid dynamical models which are much simpler and give accurate results with less modeling and computational effort . in authorspropose lumped parameter model of the bells mounted in central european system and prove that with the model we are able to predict impact acceleration and bell s period of motion .  in our previous publication we present an improved hybrid dynamical model of the yoke - bell - clapper system .all parameter values involved in the model have been determined basing on the measurements of the biggest bell in the cathedral basilica of st stanislaus kostka , lodz , poland .proposed model is validated by comparing the results of numerical simulations with experimental data .the presented results show that the introduced model is a reliable predictive tool and can be used for further case studies  in this paper we describe the most common working regimes of bells and investigate how the yoke design and the propulsion mechanism influence their dynamics . to simulate the behavior of the yoke - bell - clapper system we use the hybrid dynamical model that we present in detail in our previous publication .we characterize the solutions that can be considered as the proper operation of the bell and analyze how the geometry of the yoke and the driving motor output affect the dynamics of the system .we introduce two influencing parameters and develop diagrams that allow to asses how the bell s behavior depends on the yoke type and propulsion amplitude .such plots describe how presumed working regimes can be obtained and can be beneficial during the design and/or restoration processes of the bells .in addition , we investigate the time that is needed to reach given attractor .presented results prove that in some cases special launching procedure of the instrument should be introduced to shorten the time of transient motion .finally , we propose simple and universal control of the driving mechanism that allows to decrease transient time significantly .  the paper is organized as follows . in section 2we describe the hybrid dynamical model of the church bell and introduce parameters that influence the system s dynamics . in section 3we characterize the 7 most common working regimes and in section 4 investigate how they can be obtained .finally , in section 5 the conclusions are drawn .the hybrid dynamical model of the yoke - bell - clapper system that we consider in this paper has been described in detail in our previous publication . to develop the model anddetermine its parameters values we have performed detailed measurements of the existing bell named _`` the heart of lodz '' _ ( the biggest bell in the cathedral basilica of st stanislaus kostka in lodz ) . using the same bellwe have tuned and validated the model by comparing the results of numerical simulations with the data collected during a series of experiments . presented resultsshow that the model is a reliable predictive tool which can be used to simulate the behavior of a wide range of yoke - bell - clapper systems . in next subsectionswe briefly describe the derivation of the equations of motion and present the influencing parameters .the model that we use is build up based on the analogy between freely swinging bell and the motion of the equivalent double physical pendulum .the first pendulum has fixed axis of rotation and models the yoke together with the bell that is mounted on it .the second pendulum is attached to the first one and imitates the clapper .the photo of the bell that has been measured to obtain the parameters values is presented in fig.[fig : figure1 ] ( a ) . in figs.[fig : figure1 ] ( b , c ) we show schematic model of the bell indicating the position of the rotation axes of the bell , the clapper and presenting parameters involved in the model . for simplicity , henceforth , we use term `` bell '' with respect to the combination of the bell and it s yoke which we treat as one solid element .  ]the model involves eight physical parameters .parameter describes the distance between the rotation axis of the bell and its center of gravity ( point ) , is the distance between the rotation axis of the clapper and its center of gravity ( point ) .the distance between the bell s and the clapper s axes of rotation is given by parameter .the mass of the bell is described by parameter , while parameter characterizes the bell s moment of inertia referred to its axis of rotation .similarly , parameter describes the mass of the clapper and stands for the clapper s moment of inertia referred to its axis of rotation .considered model has two degrees of freedom . in fig .[ fig : figure1 ] ( d ) we present two generalized coordinates that we use to describe the state of the system : the angle between the bell s axis and the downward vertical is given by and the angle between the clapper s axis and downward vertical by .parameter ( see [ fig : figure1 ] ( d ) ) is used to describe the clapper to the bell impact condition which is as follows :    synonymously , collision between the bell and the clapper occurs when the absolute difference between the bell s and the clapper s angular displacements is equal to .in this section we present the mathematical model that we use to simulate oscillatory motion of the investigated yoke - bell - clapper system .we use lagrange equations of the second type and derive two coupled second order odes that describe the motion of the considered system ( full derivation can be found in ) :      where stands for gravity and describes the effects of the linear motor propulsion .the motor is active - and excites the bell - when its deflection from vertical position is smaller than  ] ,  ] ,  ] ,  ] and  ] ,  ] . as aforementioned ,all parameters values have been evaluated specifically for the purpose . for integration of the model described abovewe use the fourth - order rungekutta method .odes [ eq1 ] and [ eq2 ] together with the discreet model of impact described in the next subsection create a hybrid dynamical system .   after considering free motion conditions , in this sectionwe briefly refer to the discreet impact model which is was proposed by meneghetti and rossi and validated experimentaly in our previous publication .when the condition [ imp_con ] is fulfilled we stop the integration process .then , instead of analyzing the collision course , we restart simulation updating the initial conditions of equations [ eq1 ] and [ eq2 ] by switching the bell s and the clapper s angular velocities from the values before the impact to the ones after the impact .the angular velocities after the impact are obtained taking into account the energy dissipation and the conservation of the system s angular momentum that are expressed by the following formulas :    \dot{\varphi}_{1,bi}+\left[b_{c}+ml_{c}l\cos\left(\varphi_{2}-\varphi_{1}\right)\right]\dot{\varphi}_{2,bi}=\\ \\\left[b_{b}+ml_{c}^{2}+ml_{c}l\cos\left(\varphi_{2}-\varphi_{1}\right)\right]\dot{\varphi}_{1,ai}+\left[b_{c}+ml_{c}l\cos\left(\varphi_{2}-\varphi_{1}\right)\right]\dot{\varphi}_{2,ai } \end{array}\label{cons_mom}\ ] ]  where index stands for `` after impact '' , index for `` before impact '' and parameter is the coefficient of energy restitution . in our simulationswe assume referring to a series of experiments performed by rupp et .al . . in our previous investigation we have analyzed influence of on the response of system and we have proved that dynamics of the system barely changes for small alterations of parameter ( ) . hence , we claim that there is no need to further adjust the value of for the considered bell .the mathematical model of the ringing bell described in section [ sec : model - of - the ] contains 12 parameters ( , , , , , , , , , , , ) but most of them are self dependent .moreover , we have to remember that we consider a musical instrument , hence we can not change some of its features as it could affect the sound it generates . in real applications we can easily modify the driving motor and the mounting of the bell ( by changing the design of the yoke ) .therefore , we try to describe how changes of the propulsion and the yoke s design influence the system s dynamics . as a reference , we use values of parameters characteristic for _ `` the heart of lodz '' _ and alter them to simulate the changes in the propulsion or mounting layout .  in our investigationwe assume the linear motor propulsion that is described by piecewise function given by the formula [ eq : mt ] .hence , we can modify the effects generated by the motor by changing the range of bell s deflection in which the motor is active ( in our case ) or by altering the maximum generated torque .practically , it is much easier to modify the maximal momentum which can be realized either by replacing the motor or by changing the length of the torque arm .therefore , we take as the parameter that describes the driving motor characteristic and use it as the first controlling parameter .  to describe the modifications of the yoke we introduce new parameter and take it as the second influencing parameter . in fig .[ fig : lr ] we explain the meaning of parameter .  : ( a ) reference yoke design ( based on _ `` the heart of lodz '' _ yoke ) , ( b ) , ( c ) .[fig : lr ] ]  for the reference yoke ( _ `` the heart of lodz '' _ yoke ) we assume .if we change the yoke design so that the bell s center of gravity is lowered , then and as the value of we take the distance by which the bell s center of gravity is shifted with respect to the reference yoke .similarly , if the bell s center of gravity is elevated , we assume and take its displacement as the value of . as a maximum considered value of take  ] rotation axis of the bell goes through its center of mass .it is important to point out that changes of value affect other parameters of the model .hence , whenever we change the value of we have to swap three parameters : has to be replaced by , must be replaced by , and finally by .values of , and are calculated using the following formulas :   considered system is not only hybrid and nonlinear but also piecewise due to the model of driving linear motor .hence it can exhibit a plethora of different dynamical phenomena such as periodic , quasi periodic and chaotic attractors with different number of collisions .therefore , a particular bell can behave quite differently depending on the yoke design and the driving technique .still , only few types of periodic attractors can be considered as a proper working regimes and have practical applications .these regimes are often called ringing schemes and can be classified in groups that have common characteristics .the features that are important during the categorization are principally : the period between the consecutive impacts , the number of collisions during one period of motion and the course of the collisions . in this sectionwe point out and describe the seven most commonly meet types of behavior .we use four types of plots to present considered working regimes : phase portraits of the bell and the clapper , trajectory projected in the section of phase space showing relation between the angular position of the bell and the clapper and time trace of the clapper s velocity .we use blue lines to present attractors and trajectories ; red lines and arrows are used to present changes of velocity that are the effects of collisions while green lines are plots of the impact condition [ imp_con ] and indicate when the collisions occur .we say that the bell works in a `` falling clapper '' manner if the collisions between the bell and the clapper occur when they perform an anti - phase motion .this type of behavior is common for bells that are mounted in the european manner and _ `` the heart of lodz '' _ is an example of a bell that works in this regime . in the falling clapper ringing schemethe amplitude of the clapper s motion is smaller than the bells.the clapper s velocity sign changes when collision occurs .the amount of energy that is transferred during the impact is relatively large and sufficient to let the bell resound nicely .thanks to this feature it is relatively easy to receive nice voicing of the bells working in falling clapper regime .but , in some cases , the energy transfer is too abrupt which can lead to crack or damage of the bell or the clapper .impacts per period obtained for  ] .phase portraits of the bell ( a ) , the clapper ( b ) , section of phase space with green lines that indicate impact conditions ( c ) and time trace of the clapper s velocity ( d ) .effects of collisions ( changes of velocity ) are marked with red lines .[ fig : fall_2i ] ]  we can distinguish a symmetric type of falling clapper with collisions per one period of motion and its asymmetric version with impact per period .although the course of the collision is similar in both cases they are completely different for the listeners as the time interval between successive impacts is almost doubled in the asymmetric case . in figs .[ fig : fall_2i ] and [ fig : fall_1i ] we present two types of falling clapper ringing scheme .impact per period obtained for  ] .phase portraits of the bell ( a ) , the clapper ( b ) , section of phase space with green lines that indicate impact conditions ( c ) and time trace of the clapper s velocity ( d ). effects of collisions ( changes of velocity ) are marked with red lines.[fig : fall_1i ] ]   if the collisions between the bell and the clapper occur when they perform in - phase motion we say that the bell works in a `` flying clapper '' manner. in this regime the amplitude of the clapper s motion is larger than the bell s and the clapper s velocity sign remains the same after the collisions .the collisions have more gentle characteristic and the amount of energy that is transferred during the impact is much smaller than in `` falling clapper '' ringing scheme .therefore , sometimes it may be difficult to achieve nice resounding of the bell , but the risk of the crack or damage is significantly decreased . in figs .[ fig : fly_2i ] and [ fig : fly_1i ] we present two types of `` flying clapper '' behavior : symmetric attractor with impacts per period and asymmetric one with only impact per period respectively.   impacts per period obtained for  ] .phase portraits of the bell ( a ) , the clapper ( b ) , section of phase space with green lines that indicate impact conditions ( c ) and time trace of the clapper s velocity ( d ) .effects of collisions ( changes of velocity ) are marked with red lines.[fig : fly_2i ] ]   impact per period obtained for  ] .phase portraits of the bell ( a ) , the clapper ( b ) , section of phase space with green lines that indicate impact conditions ( c ) and time trace of the clapper s velocity ( d ) .effects of collisions ( changes of velocity ) are marked with red lines.[fig : fly_1i ] ]   the `` double kiss '' is the name of working regime in which we observe impacts per one period of motion . during one periodthe clapper hits each side of the bell s shell twice .the first collision on each side is in the `` falling clapper '' manner while the second impact has `` flying clapper '' course .this behavior is especially attractive for the listeners .still , it is rarely met mainly because it is difficult to achieve . in the next sectionwe precisely determine conditions ( ranges of parameters and ) for which we can observe this working regime . in fig .[ fig : double_kis ] we show the characteristics of `` double kiss '' .impacts per period .the plots were obtained for  ] .phase portraits of the bell ( a ) , the clapper ( b ) , section of phase space with green lines that indicate impact conditions ( c ) and time trace of the clapper s velocity ( d ) .effects of collisions ( changes of velocity ) are marked with red lines.[fig : double_kis ] ]`` sticking clapper '' is the name of working regime in which the clapper and the bell remain in contact for a certain amount of time . in other wordsthe `` sticking clapper '' refers to the attractors that contain sliding mode .this working regime is typical for the bells mounted and operated in the english manner . in the considered systemprior the sliding mode we observe a number of successive impacts ( usually ) that have a `` falling clapper '' course . the energy amount that is transferred between the bell and the clapper decreases with each subsequent collision .hence , the sound effects caused by each hit are different and not all collisions may be noticed by the listener . moreover , when the clapper remains in the contact with the bell s shell it also influences the produced sound .all these features have to be taken into consideration if we want the bell to sound nicely . ] .phase portraits of the bell ( a ) , the clapper ( b ) , section of phase space with red lines that indicate impact conditions ( c ) and time trace of the clapper s velocity ( d ) .effects of collisions ( changes of velocity ) are marked with red lines.[fig : slide ] ]   if the forcing amplitude is not sufficient or the yoke is designed improperly , we can observe stable periodic attractor with no collisions .in such conditions no sound is produced and the bell can not work as a musical instrument .unfortunately , no impacting attractors occur in wide range of and and even nowadays it is common that we have to redesign the yoke to make ringing possible .apart from the ringing schemes described in this section we can indicate other periodic attractors that can be successfully employed . for example we can observe asymmetric flying clapper behavior with doubled period and impacts per period . such behavior can be easily taken as a typical falling clapper .similarly , a quasi - periodic attractor with almost equal time intervals between the subsequent impacts can sound almost like a periodic ringing scheme .moreover , as we proved in our previous publication , the course of successive collisions can differ a bit .hence , in practical applications we do not demand strictly periodic behavior .instead , we want to receive similar collisions of a presumed type in a possibly equal time intervals . still , when designing the yoke and the propulsion of the bell we should always look for conditions that ensure periodic behavior as it increases the probability that the real system will work properly and reliably .in fig . [ fig : lr-1a ] we present one parameter diagram that presents how the system behaves with varying value of parameter .we assume  ] .each time we start integration form zero initial conditions ( , , , ) , hence it is not a bifurcation diagram ( the bell and the clapper always start their motion from hanging down position with zero velocities ) .we investigate the behavior of the system after minutes of preliminary motion ( the analysis of influence of transient time on the system s dynamics is shown in the end of this section ) . to determine in which regime the system operates , we use two indicators .the first is the absolute value of the clapper s angular velocity when the clapper passes the downward vertical .the second is the clapper s velocity just after the clapper to the bell impact .moreover , we calculate the energy dissipated at instant of collision .we record these values during the period of two minutes .subplots ( a ) and ( b ) of fig .[ fig : lr-1a ] correspond to the first and the second indicator respectively while in lower part of the figure ( subplot ( d ) ) we present the energy dissipation . analyzing fig .[ fig : lr-1a ] ( a , b ) we can asses the periodicity and the symmetry of the attractor. moreover , we are able to evaluate the number of collisions per period of motion . to determine if we observe the `` flying clapper '' or `` falling clapper '' ringing scheme , we have to look at the phase portrait of the clapper or at least value of the clapper angular velocity just before the impact .we focus on the clapper s motion because the changes in the bell s behavior are often imperceptible due to its much bigger mass and inertia .supblot ( a ) presents clapper s angular velocity when the clapper passes the downward vertical , ( b ) shows clapper s velocity just after the impact and subplot ( c ) presents the energy dissipated at instant of collision .[ fig : lr-1a ] ]  the two most common ringing schemes are symmetric `` flying '' and `` falling '' clapper . analysing subplot ( c ) of fig .[ fig : lr-1a ] we see that in `` flying clapper '' working scheme collisions have gentle course and the risk of damage of the bell or the clapper is minimal . in this regimethe amount of energy dissipated during collision is relatively small and do not depend on the yoke design .conversely in `` falling clapper '' ringing scheme the amount of energy dissipated at instant of collision strongly depends on the yoke design .collisions are more abrupt but still the bell can operate reliably if the and parameters are properly chosen .the risk of failure is also higher for asymmetric attractors ( both `` falling '' and `` flying '' clapper ) because then the amount of energy transferred during collisions is significantly higher than in symmetric case ( for similar parameters values ) .that is why these working regimes are rarely met .vertical lines in fig .[ fig : lr-1a ] mark the ranges of where given working regimes can be observed .we see that transitions between different states of the system are sudden and sometimes even minor changes in the yoke geometry can result in a completely different behavior of the system .we usually expect the bells to work reliably for decades without any maintenance .therefore , the yoke and the propulsion of the bell should be designed to ensure that the instrument will work properly regardless of small changes of influencing parameters .such approach requires the in - depth knowledge of the systems dynamics .   to examine how the yoke s design - described by parameter - and the amplitude of forcing - parameter - influence the system s dynamicswe perform the series of numerical simulations . in our analysiswe consider the following ranges of these parameters :  ] .we take equally spaced values of ( with the step  ] to  ] ) . that gives us different sets of system s parameters .for each one we simulate the system s motion starting from zero initial conditions ( , , , ) .  in figs .[ fig : wykres1 ] and [ fig : wykres38 ] we show the possible behavior of the systems minute and minutes after the excitation started respectively .we call these two parameters plots as the `` ringing scheme diagrams '' because they allow to determine which ringing scheme can be achieved for the given values of and .we consider most characteristic types of the bells behavior that are described in detail in the previous section .of course many different stable attractors can exist for a given set of parameters and , but during our investigation we concern only the solutions that basins of attraction contain zero initial conditions ( , , , ) .as we mention before , we do so because in most cases the bell and the clapper start their motion from hanging down position with zero velocities .another aspect that we consider is the transient time needed to reach a given attractor .it is important in real applications because the launching procedure of the bell should be possibly short .hence , when designing the yoke and the propulsion mechanism we have to know both how presumed ringing scheme can be achieved as well as the time of transient behavior before we reach that attractor .if this time is too long we can either change the yoke design or apply special starting procedure - by the control of the driving motor .minutes after the excitation started .[ fig : wykres38 ] ]  the first plot - fig .[ fig : wykres38 ] - presents the state of the system minutes after the propulsion starts .it shows which stable attractor will be achieved starting from zero initial conditions for different values of and .hence , fig .[ fig : wykres38 ] indicates which ringing scheme can be obtained with given type of yoke and maximal driving torque .analyzing the plot we can draw the following conclusions .the biggest areas correspond to the three most common ringing schemes : falling clapper ( black ) and flying clapper ( red ) with 2 impacts per one period of motion and sticking clapper ( pink ) that is typical for the bells mounted in the english manner .this means that these behaviors are relatively easy to achieve and remain even after long period of time when values of some parameters can change a bit ( for example maximum torque generated by the linear motor ) .generally , the design of the yoke determines how the bell will operate . for ]we can not obtain any of the analyzed ringing schemes no matter how big the forcing amplitude is .this proves that when the yoke is designed improperly it may be impossible to force the system to ensure proper operation . for ] and do not demand to reach the stable solution .instead , we check if for the average observer the behavior of the system could be seen as proper and regular .for example we do not care if the amplitude of the bell s motion is still slightly rising and just check if the period between the successive impacts is constant and the collisions have recurrent course . comparing figs .[ fig : wykres38 ] and [ fig : wykres1 ] we see that the difference in the size of areas is especially visible for  ] the areas corresponding to different ringing schemes are almost identical to the ones presented in fig .[ fig : wykres38 ] .the main difference is that for  ] and impacts per period for  ] ,  ] it is equal to parameter . hence , there is no jump in the torque generated by the motor when the launching procedure ends (  ] and $ ] ) changes during the analyzed first minutes of the system s motion . the dotted vertical line in fig .[ fig : max_torque ] marks the time when the launching procedure ends and the output of the motor stabilizes .minutes after the excitation started ) .[ fig : wykres_control ] ]  in fig .[ fig : wykres_control ] we present the ringing scheme diagram obtained for the system with applied launching procedure that is described above .similarly to fig .[ fig : wykres1 ] , this time we also do not require the system to reach the stable attractor and just check if its behavior could be perceived as proper by an average listener . analyzing fig .[ fig : wykres_control ] we see that even using simple control of the driving motor we can achieve satisfactory results and decrease the time of the transient motion .moreover , obtained results prove that formula [ eq : mt - control ] - universal for all and - allow us to achieve good results in a wide range of parameters values .but , the proposed control is not efficient with respect to the systems with large forcing amplitude and yoke design that provides large inertia of the bell ( left upper part of the plot . ). comparing figs .[ fig : wykres38 ] and [ fig : wykres_control ] we see the difference in the borders and sizes of the areas corresponding to different working regimes , but still thanks to the proposed launching procedure we can reach all considered working regimes .hence , the proposed control allow us to obtain all considered ringing schemes but they appear in smaller ranges of parameters values .therefore , in some cases there is a need to adjust the launching procedure to shorten transient time and reach presumed behavior of the system .hybrid dynamical model of the yoke - bell - clapper system can exhibit a plethora of different dynamical behaviors but only a few of them can be considered as a proper working regimes of the instrument .therefore , the yoke of the bell and its propulsion mechanism should be designed to ensure the system will behave correctly and sound nicely . in case of bells with typical shape and mass engineers usually design yokes basing on their experience and local tradition .still , it is often a challenge to create a reliable suspension for unique bells and sometimes the mounting of the bell or its propulsion have to be redesigned to achieve presumed ringing scheme .  in the paperwe present a method that can be used to determine the conditions under which given type of behavior can be achieved .we develop two parameter ringing scheme diagrams that describe how the geometry of the yoke and maximum output of the driving motor influence the dynamics of the system .similar charts can be calculated for any bell and used to design its mounting and propulsion .thus , we can ensure that the instrument will work properly and reliably regardless of small changes of parameters .moreover , analyzing the ringing scheme diagrams we see that it is possible to design the yoke and the propulsion to enable more than one ringing scheme with relatively small changes of motor and/or yoke parameters .hence , we can easily differentiate the tune of the bell for different occasions .basing on the presented analysis we are able to determine the minimum required power of the driving motor and estimate the time that is needed to achieve given attractor .if the transient time is too long , special launching procedure has to be applied .we propose the control of the driving motor that enables to reach all considered ringing schemes after seconds of preliminary motion .it is possible to further optimize the launching procedure , but to get best results we need to know the values of all system s parameters .still , presented results are robust and prove that even simple and universal formula can be highly efficient in a wide range of systems parameters .this work is funded by the national science center poland based on the decision number dec-2013/09/n / st8/04343 .we would especially like to thank the parson of cathedral basilica of st stanislaus kostka prelate ireneusz kulesza for his support and unlimited access to the bell .we have been able to measure the bell s template thanks to the bell s founder mr zbigniew l. felczynski .the data on the clapper , the yoke and the motor have been obtained form mr .pawel szydlak .18   s. ivorra and j. r. cervera .analysis of the dynamic actions when bells are swinging on the bell - tower of bonrepos i mirambell church ( valencia , spain ) ._ iproc . of the 3rd international seminar of historical constructions _ , pages 41319 , 2001 .j. klemenc , a. rupp , and m. fajdiga .a study of the dynamics of a clapper - to - bell impact with the application of a simplified finite - element model ._ engineering with computers _ , 270 ( 3):0 261272 , 2011 .r. spielmann , a. rupp , m. fajdiga , and b. aztori .kirchenglocken - kulturgut , musikinstrumente und hochbeanspruchte komponenten ._ glocken - lebendige klangzeugen .schweizerische eidgenossenschaft , bundesamt fur kultur bak ._ , pages 2239 , 2008 .

Abstract: in this paper we investigate dynamics of church bells , characterize their most common working regimes and investigate how to obtain them . to simulate the behavior of the yoke - bell - clapper system we use experimentally validated hybrid dynamical model developed basing on the detailed measurements of the biggest bell in the cathedral basilica of st stanislaus kostka , lodz , poland . we introduce two parameters that describes the yoke design and the propulsion mechanism and analyze their influence on the systems dynamics . we develop two - parameter diagrams that allow to asses conditions that ensures proper and smooth operation of the bell . similar charts can be calculated for any existing or non - existing bell and used when designing its mounting and propulsion . moreover , we propose simple and universal launching procedure that allows to decrease the time that is needed to reach given attractor . presented results are robust and indicate methods to increase the chance that the instrument will operate properly and reliably regardless of changes in working conditions .  bells , dynamics , impacting system , hybrid system , ringing schemes


the main title of this paper is , apart from the term multiplex , literally identical to a forney , gallager and co - workers paper issued more than one decade ago [ for et al .84 ] , which analyzed the benefits of coded - modulation techniques .the large success achieved by ungerboeck s coded - modulation came from the way of introducing redundancy in the encoder [ ung 82 ] . in classical channel coding ,redundant signals are appended to information symbols in a way somewhat analogous to time division multiplex tdm ( envelope interleaving ) .it was believed that introducing error - control ability would increase bandwidth .an efficient way of introducing such an ability without sacrificing rate nor requiring more bandwidth consists in adding redundancy by an alphabet expansion .this technique is particularly suitable for channel in the narrow - band region .a similar reasoning occurs in the multiplex framework where many people nowadays believe that mux must increase bandwidth requirements .+  + one of the most powerful tools in communications is the fourier transform , specially its discrete version dft . on the other side ,applications of finite field have a renewed explosion of interest in the last decades .a discrete fourier transform for finite fields ( ffft ) was introduced by pollard [ pol 71 ] .it has successfully applied to perform discrete convolution and as a tool of image processing [ ree et al .77 , ree - tru 79 ] among many other applications .besides classical ffft we are concerned with a new finite field discrete version [ cam et al .98 ] of the integral transform introduced by r.v.l .hartley [ har 42 , bri 92 ] .alike classical galois - fourier transforms [ bla 79 ] , finite field hartley transform ght [ cam et al .98 ] defined on a gaussian integer set gi( ) contains some redundancy and only the cyclotomic coset leaders of transform coefficients are needed to be transmitted .this yields a new * efficient - bandwidth code division multiplex for band - limited channels*. these mux may present lower bandwidth requirements than tdm by using an argument somewhat analogous to coded - modulation .tradeoffs between extension of the alphabet and bandwidth are exploited in the sequel .+  + in the present work , the coded - modulation idea is adapted to multiplex : information streaming from users are not combined by interleaving ( like tdm ) but rather by signal alphabet expansion .the mux of users sources over a galois field gf( ) deals with an expanded signal set having symbols from an extension field gf( ) , . as a consequence ,the multiplex of band limited channels of identical maximal frequency leads to _ bandwidth requirements less than _ , in contrast with tdmed or fdmed signals .+  + the design of such an _ efficient bandwidth mux _ is based upon the finite field structure , specifically , it consists of applying finite - field fourier transforms ( ffft ) .this paper shows that the bandwidth compactness factor relatively to tdm depends on the length , the alphabet extension .+  + another point to mention is that the superiority of digital mux regarding analogic mux is essentially due to the low complexity of tdm .the majority of today multiplex systems follows plesiocrhonous ( pdh ) or synchronous ( sdh ) hierarchy . besides presenting higher spectral efficiency ( bits / s / hz ) than classical mux, the new cdm schemes here introduced are based on fast transforms so they also seem to be attractive from the implementation viewpoint .although most mux systems today be intended to optical fiber which are not yet bandlimited channels , multiplex has also been adopted on satellite channels .probably applications of such multiplex will be on transponders and specially on cellular communications .cable television ( catv ) can also have benefit by using such a technique .digital multiplex normally alludes time division multiplex ( tdm ) .however , it also can be achieved by coding division multiplex ( cdm ) .the cdm has recently been focus of interest , specially after the is-154 standardization of the cdma system for cellular telephone . in this sectionwe introduce a new class of mux schemes based upon finite - field transforms which can be implemented by fast transform algorithms ._ classical multiplex increases simultaneously the transmission rate and the bandwidth by the same factor _ , keeping thus the spectral efficiency unchanged . in order to achieve ( slight ) better spectral efficiencies ,classical cdma uses waveforms presenting a non zero but residual correlation .+  + given a signal over a finite field gf( ) we deal with the galois domain considering the spectrum over an extension field gf( ) which corresponds to the finite field transform ( galois transform ) [ bla 79 , cam et al .+  + as an alternative and attractive implementation , the multiplex is carried out by a finite field transform ( ffft / ffht ) and the demux corresponds exactly to a inverse finite - field transform of length .each symbol on the ground field gf( ) have duration seconds .an -user mux can be designed on the extension field gf( ) where . for the sake of simplicity , we begin with =1 and consider a ( )-channel mux as follows .typically , we can consider gf(3 ) corresponding to alternate mark inversion ami signaling .    a ( )-cd consider digital carrier sequences per channel as versions of cas function over the galois ( complex ) field gi( ) . the cas ( cos and sin ) function is defined in terms of finite field trigonometric functions [ cam et al .98 ] , .+  + carrier 0 : + + carrier 1 : + + ... + carrier : + + ... + carrier : + .+  + the cyclic digital carrier has the same duration of a input modulation symbol , so that it carries slots per data symbol .the interval of each cas - symbol is and therefore the bandwidth expansion factor by multiplexing channels may be roughly , the same result as fdm and tdm / pam . + a first scheme of the multiplex is showed : the output corresponds exactly to the galois - hartley transform of the `` users''-vector .+ therefore , it contains all the information of the channels .each coefficient of the spectrum has duration .these carriers can also be viewed as spreading waveforms .an -user mux has spreading sequences , one per channel .the requirements to achieve welch s lower bound according to massey and mittelholzer [ mas - mit 91 ] are hold by sequences .the matrix $ ] presents both orthogonal rows and columns and have the same energy. +  + a naive example is present in order to illustrate such an approach .a 4-channel mux over gf(5 ) can be easily implemented .it is straightforward to see that such signal are not fdmed nor tdmed .hlio magalhes de oliveira was born in arcoverde , pernambuco , brazil , in 1959 .he received both the b.sc . anddegrees in electrical engineering from universidade federal de pernambuco ( ufpe ) , recife , pernambuco , in 1989 and 1983 , respectively .then he joined the staff of the same university as a lecturer . in 1992he earned the docteur de l__cole nationale suprieure des tlcommunications _ _ degree , in paris . he is currently with the statistics departement of ufpede oliveira was appointed as honored professor by fivety electrical engineering undergratuate classes and chosen as the godfather of sixteen engineering graduation .his publications are available at arxiv .research current interests include : applied statistics , communications theory , applied information theory , signal analysis , and wavelets .

Abstract: a new efficient - bandwidth code - division - multiple - access ( cdma ) for band - limited channels is introduced which is based on finite field transforms . a multilevel code division multiplex exploits orthogonality properties of nonbinary sequences defined over a complex finite field . galois - fourier transforms contain some redundancy and just cyclotomic coefficients are needed to be transmitted yielding compact spectrum requirements . the primary advantage of such schemes regarding classical multiplex is their better spectral efficiency . this paper estimates the _ bandwidth compactness factor _ relatively to time division multiple access tdma showing that it strongly depends on the alphabet extension . these multiplex schemes termed galois division multiplex ( gdm ) are based on transforms for which there exists fast algorithms . they are also convenient from the implementation viewpoint since they can be implemented by a digital signal processor .   digital multiplex , code - division multiple access , hartley - galois transform , finite field transforms , spread sequence design .


complex systems can be represented as networks , where the main units of the system become nodes and interacting units are connected by edges .the last years have witnessed an intense research activity on networks by the scientific community , after the discovery that many systems in nature , society and technology , turn into graphs with peculiar properties  .in particular , many networks are characterized by a heterogeneous distribution of the number of neighbors of a node , or degree , where nodes with low degree coexist with nodes with large degree ( hubs ) .such heterogeneity is responsible for a number of remarkable features of real networks , such as resilience to random failures / attacks  , and the absence of a threshold for percolation  and epidemic spreading  .the presence of nodes with different degrees means that there is a broad diversification of their roles within the graph .centrality measures are designed to rank graph nodes based on their topological importance  . among the most popular centrality measures we mentiondegree itself , but also measures depending on shortest paths between nodes pairs , like node betweenness and closeness .there are as well centrality measures that depend on spectral properties of graph matrices .these measures are important because they are usually associated to simple dynamic processes taking place on graphs , like diffusion .in particular , the pagerank algorithm , proposed by the google founders brin and page  , managed to turn google into the leading interface between users and the world wide web . in recent work spectral properties of graph matrices have also been used to characterize the participation of nodes in network subgraphs ( subgraph centrality )  and to estimate the bipartitivity of graphs  .however , spectral centrality measures have not been much investigated in the physics literature .we shall introduce and review four centrality measures : pagerank , eigenvector centrality  and the hub / authority scores introduced by kleinberg for his hits algorithm  .these measures are usually adopted on directed graphs , we shall as well discuss extensions to the undirected case , where applicable .in section [ sec1 ] we present the measures and describe them in some detail .analytical and numerical results on particular graphs will be shown in section [ sec2 ] , whereas in section [ sec3 ] we shall compare the rankings of nodes of real graphs for different centrality measures .conclusions will be reported in section [ sec4 ] .the basic matrix of a graph is the _ adjacency matrix _ , where the element equals , if nodes and are connected by a link , if they are not .if the network is directed , the adjacency matrix is not symmetric . in this case, it is necessary to distinguish between two types of links adjacent to a node , i. e. links pointing to the node ( incoming ) and links pointing outside ( outgoing ) .therefore , there are two types of degree : _ indegree _ , i.e. the number of incoming links ; _ outdegree _ , i.e. the number of outgoing links .likewise , one distinguishes between the _ in - neighbors _ of a node , i.e. the nodes pointing at the node , and the _ out - neighbors _ , i.e. the nodes pointed at by the node .the directedness of the links has a number of important implications , involving both some basic structural concepts , like connectivity , and processes taking place on the network .for instance , a random walk is a stationary process on any undirected graph , but it is not in general on a directed graph , due to the possible presence of _ dangling ends _ , i.e. nodes with zero outdegree , that act as sinks for the process . on the other hand, diffusion leads to a natural definition of centrality , based on the frequency that a walker stops by a node during the process . in order to make a diffusive process stationary on a directed graph , one needs to give the walker the opportunity to leave from a dangling end .pagerank offers a simple solution , which we describe below .pagerank ( pr ) is the prestige measure used by google to rank web pages .it is supposed to simulate the behavior of a user browsing the web .most of the times , the user visits pages just by surfing , i.e. by clicking on hyperlinks of the page he is on ; otherwise , the user will jump to another page by typing its url on the browser , or going to a bookmark , etc .. on a graph , this process can be modelled by a simple combination of a random walk with occasional jumps towards randomly selected nodes .this can be described by the simple set of implicit relations here , is the number of nodes of the graph , is the pr - value of node , the outdegree of node and the sum runs over the nodes pointing towards .the _ damping factor _ is a probability , that weighs the mixture between random walk and random jump . on practical applicationsit is usually set to small values ( typically 0.15 ) . for any process reaches stationarity , as a walker has a finite ( no matter how small ) probability to escape from a dangling end , whenever it lands there . when , the process may not be stationary and pr is ill defined .when , instead , the jumping process dominates and all nodes have the same pr - value .pr goes beyond indegree : in order to have a large pr - value for a node it is important to have many neighbors pointing at a node , i.e. large indegree , but it is also important that the neighbors have large pr - values .so , if two nodes have equal indegree , the node with more `` important '' neighbors will have larger pr .solving the set of equations ( [ eq1 ] ) is equivalent to solving the eigenvalue problem for the transition matrix , whose element is given by the following expression : pr is just the principal eigenvector of , and is usually determined with the power method , i.e. by repeatedly multiplying the matrix by an arbitrary vector until all the entries of the resulting vector are stable .this is also the procedure we adopted to compute the eigenvectors corresponding to all centrality measures we studied .the literature on pr is very large , because of its huge impact on web search . in one of the first theoretical studies , the dependence of pr on the damping factor was investigated . in general , the attention has been mostly focused on the graph of the world wide web , where web pages are nodes and the hyperlinks their connections .comparatively little has been done to study the measure on more general classes of networks .a recent mean field study  has shown that the average pr value of nodes with the same indegree is a linear function of indegree in the absence of degree - degree correlations . in another study ,some analytical results were found on pr distributions on special classes of graphs  . in section [ subsec3.1 ]we shall briefly resume the results of  and build up on them .the eigenvector centrality ( ev ) is also based on the principle that the importance of a node depends on the importance of its neighbors . in this casethe relationship is more straightforward than for pr : the prestige of node is just proportional to the sum of the prestiges of the neighboring nodes pointing to it from eq .( [ eq3 ] ) we see that is just the -component of the eigenvector of the transpose of the adjacency matrix with eigenvalue .we notice that the trivial eigenvector with all components equal to zero is always a solution of eq .( [ eq3 ] ) .the true ev is then associated to the existence of non - trivial solutions of the eigenvalue problem of eq .( [ eq3 ] ) . from eq .( [ eq3 ] ) we see that nodes with indegree zero also have zero centrality : in general , nodes pointed at by nodes with zero centrality also have zero centrality and this effect will propagate to other nodes , so that in many cases ev would not give any information about a big number of nodes . to avoid this , it is useful to make the following modification : to each node we assign a prestige , which is independent of its relationships with the other nodes .( [ eq3 ] ) is then modified as follows : the role of the parameter reminds that of the damping factor in pr .the parameter weighs the relative importance of the contribution of the peers versus that of the node itself . the new measure is called _ -centrality _ ( )  and is the one we shall investigate in this paper .we remark that , in contrast to pr , here the solutions do not have a natural interpretation in terms of probability , so the sum of the -centralities need not be .however we shall normalize the final values by dividing them by their sum , so to make them add up to , for practical purposes .google s pr was not the first prestige measure for web pages based on the web s graph representation . shortly before the seminal paper by brin and page , jon kleinberg had proposed another solution to the problem of ranking web sites based on their importance for the users .this solution was the _ hits algorithm _ , which distinguishes two types of web pages : _ hubs _ and _ authorities_. let us suppose that a user submits a query through a search engine . if a page is very relevant for this query , one can reasonably expect that it will be pointed at by many other pages .however , the simple indegree would not allow to discriminate the relevant pages from other pages with similar ( large ) indegree .an important difference is that pages pointing to a relevant page are likely to point as well to other relevant pages , so to create a sort of bipartite structure where relevant pages ( authorities ) are cited by special pages / indices ( hubs ) .such bipartite structures allow to identify the relevant pages for the user query .therefore one assigns two scores to a page of the web : the _ hub score _ and the _ authority score _ .pages with high authority scores are pointed at by pages with high hub scores . in turn , a good hub points at ( very ) authoritative pages .this mutually reinforcing mechanism is described by the coupled relations which can be rewritten in the form of simple eigenvalue equations for both and by substitution  from eqs .( [ eq7 ] ) and ( [ eq8 ] ) we see that the hub and authority scores are just eigenvectors of the matrices and .we stress that both and are symmetric , whether is symmetric or not .the scores and correspond to the principal eigenvectors of these matrices .in  the two main limits of the pr measure , corresponding to and , were investigated .analytical results can be derived for special graphs , such as graphs grown with popular mechanisms , like preferential attachment  . for our proofswe shall focus on the model by dorogovtsev , mendes and samukhin ( dms )  , which generates graphs with power - law degree distributions with any exponent larger than . in this model , at each time step a new node is added to the graph and links are set from the new node to preexisting ones .the probability that a new node gets attached to a node ( with indegree ) is where is a positive constant .when one recovers the recipe of the original preferential attachment formulation of barabsi and albert  . in general , the exponent of the indegree distribution .for simplicity , we shall study the special case in which , i.e. each node has outdegree and the network is a tree .the results are however independent of .we assume that is very small . to the first order in , and remembering that each node has outdegree by construction , eq .( [ eq1 ] ) takes the following form which looks particularly simple , though not generally solvable . from eq .( [ eq10 ] ) we see that the pr of a node equals a constant plus the pr of its in - neighbors .this recipe enables to calculate pr recursively on simple trees , as shown in fig .[ fig1 ] , where we focus on a subgraph of a tree .subgraph of a tree .the pr - values of all nodes shown can be simply calculated.,width=188 ]  node is the root of the subgraph as every walk starting on any of the nodes will reach at some stage .we call any node with this property a _ predecessor _ of . the pr value of any node of the graph is determined only by its predecessors. in the case illustrated , the calculation is particularly simple : we start from the leaves of the subgraph ( empty circles ) whose pr is just because they have no incoming links , and move towards . for each node, we apply the relation ( [ eq10 ] ) .the final values are reported next to the nodes . from this examplewe can deduce a number of general properties :  * all pr values are multiples of the elementary unit ; * pr increases if one moves from a node to another by following a link ; * the pr of each node , in units of , equals the number of its predecessors .since pr takes only discrete values , in the following we shall measure it in units of .we thus indicate the distribution with , with .  in a dynamic process like network growth ,it is crucial to see what happens to the pr values / distribution when a new node comes into the picture .this is shown in fig .[ fig2 ] , where a new node is added to the network of fig . [ fig1 ] .we see that only the nodes encountered along the path from to , including , are affected , while the others retain their pr values .in particular the presence of the node determines an increase by in the pr values of the affected nodes .if a new node n gets attached to any node of the subgraph , it adds an equal contribution to the pr of all nodes in a path from to the root.,width=283 ]  now we are ready to build a master equation for the pr distribution on a dms graph . at time , the graph has nodes and links ( the root does not generate links ) ; the pr distribution is . if we add node we get a new distribution . as we have seen above, the new node will contribute an additional to the pr of the nodes in the path from to the root of the graph .we need to compute the balance between the nodes passing from pr to and those passing from to .the probability that the pr of node , initially equal to , will be changed by the new node equals the probability that the link set by the new node gets attached to one of the predecessors of ( including ) and equals where means that is a predecessor of .none of the predecessors of , other than can reach pr because of the new node , as their initial values are necessarily smaller than .the number of predecessors of ( including ) is and the total number of adjacent links to the predecessors is ( one for each predecessor , except ) .so , the number of nodes with pr that are affected by the presence of the new node and its link is then and the master equation reads eq .( [ eq14 ] ) holds for . for modification is necessary , as there can not be nodes with zero pr , so the term is not defined .however , since the new node has no incoming links , the number of nodes with pr increases by because of the new node , so we can write the stationarity condition of eqs .( [ eq14 ] ) and ( [ eq15 ] ) , in the limit of large leads to the relations which has the solution [(a+1)l-1]}\sim \frac{1}{l^2 } , \mbox{for }. \label{eq17}\ ] ] we see that the pr distribution in the limit on a dms tree is a power law with exponent , for any value of the parameter , including the limit case , when the indegree distribution becomes exponential .this result is confirmed by numerical simulations ( fig .[ fig3 ] ) , which also show that the hypothesis of the tree is not necessary , as long as each node has the same outdegree .  in  the same result was found for other models of network growth , like barabsi - albert preferential attachment  and the copying model  .pr distribution for small on a dms graph with nodes , and . in this casethe indegree distribution is a power law with exponent .,width=302 ]  it is possible that this property holds for general graphs where the flows converge towards a central root ( sink ). indeed , our finding agrees with the more general result on the size distribution of supercritical trees  .moreover , numerical studies have shown that the same behavior holds for the graph of internet , when one considers the distribution of the size of the basin connected to a given point  .indeed , our calculation follows the same procedure usually adopted for the calculation of the area of basins in river networks .   the case is well defined , but trivial , as all nodes end up having the same pr - value .we ask how this limit is reached . if , the contribution to pr given by the in - neighbors of a node is very small compared to the constant term , which is close to . in order to study the behavior of this term, we define the _ reduced pagerank _ of a node as we assume that all nodes have the same outdegree . in this case , to leading order in the infinitesimal eq . ( [ eq1 ] ) can be rewritten as where is the indegree of .we conclude that on any graph , the reduced pr of a node in the limit is proportional to the indegree of the node , if all nodes have the same outdegree .this result has been derived independently in  . as a consequence of eq .( [ eq19 ] ) , the distribution of the reduced pr for has the same trend as that of indegree , which can be easily verified numerically ( fig.[fig4 ] ) .reduced pr distribution for on a dms graph with nodes , and .the curve matches the indegree distribution.,width=302 ]   pr can be easily extended to undirected graphs as well .the corresponding equation reads where now is the degree of node . for the purposes of a random walk ,undirected links can be crossed in both directions , so a pure random walk now always reaches stationarity due to the absence of dangling ends .in fact , the stationary probability of a random walk on a node of any undirected graph is simply proportional to the degree of the node  .however , in eq .( [ eq20 ] ) we have still the contribution of random jumping , and it turns out that the mixed process is still hard to solve .we are not aware of a general solution in this case . in the limit pr is now well behaved , and its distribution coincides with the degree distribution of the graph . in fig .[ fig5 ] we show the distributions of reduced pr for different values of on a dms graph with a power law degree distribution and exponent .the reduced pr expresses the contribution to pr given by the random walk .we see that the curves follow the decay of the degree distribution for any value of . we have computed the reduced pr distribution on many other graphs and in all cases we found that they follow the same trend as the degree distribution .for example , in fig .[ fig5a ] we show the comparison between reduced pr and degree for a sample of the web link graph . herethe nodes are web pages of the domain .gov and two pages are connected if there is a hyperlink from one to the other .there are nodes and links .the graph is directed but pr was calculated by neglecting the directedness of the links . as we can see , the decay of the distributions of reduced pr resembles that of the degree distribution .the graph at hand is not simple like the dms networks , as it presents a large number of loops and community structure .therefore the result is likely to be general .we can show this with a simple argument .the general equation for reduced pr on undirected graphs is :    that we can solve formally by successive iteration , obtaining the general form where indicates the neighbors of the -shell of the node ; so , indicates the nearest neighbors of , the next - to - nearest neighbors , and so on . the last sum in the first line of eq .( [ eq22 ] ) is , for a given node , a sum over its neighbors .this sum , that we call , contains terms , being the degree of node .the sum can be approximated as the product , where is the expected value of the average of over the neighbors of a node of the network . in general , , where is a random variable with mean zero . in this way , it is easy to see from eq .( [ eq22 ] ) that , for any value of , the product of sums reduces to plus the sum of many random variables like . due to the central limit theorem , the latter sum , if it includes a large number of terms , yields a very small value with large probability .we can then conclude that , for sufficiently large , each term of the series in eq .( [ eq22 ] ) is proportional to with good approximation , therefore is also proportional to , for any value of the damping factor .we have verified numerically that this assertion is true for many graphs and degree distributions , without finding exceptions .   reduced pr on undirected graphs .variability of reduced pr distribution with on a dms graph with nodes , and .the degree distribution has a power law tail with exponent .,width=294 ]   reduced pr on undirected graphs .variability of reduced pr distribution with on the domain .gov of the world wide web .the degree distribution has a tail which follows fairly well a power law with exponent . to better show the agreement we have shifted the curves such that the tails overlap.,width=294 ]    the defining eq .( [ eq4 ] ) is formally analogous to eq .( [ eq10 ] ) .the only difference is that the eigenvalue is not as for pr .however , the results of section [ subsubsec3.1 ] hold as well when the outdegree is greater than ( as long as it is the same for all nodes ) , and in this case the sum of eq .( [ eq10 ] ) would include a multiplicative factor , which makes it identical to eq .( [ eq4 ] ) .we then deduce that all results found for pr in the limit hold for . herethe results are more general , because we did not need to make any approximation to get to eq .( [ eq4 ] ) as we instead needed to derive eq .( [ eq10 ] ) .in particular , it is not necessary that be very small and the nodes need not have the same outdegree , although this is the case for the graphs we considered .we conclude that the distribution of on dms graphs has a power law tail with exponent ( fig .[ fig6 ] ) .the same holds for graphs built using preferential attachment and the copying model , just as it happens for pr in the limit .distribution of on a directed dms graph with nodes , and .the dashed line indicates the predicted slope.,width=302 ]   on undirected graphs , eq .( [ eq4 ] ) becomes since .so , the of a node is proportional to the sum of the of its neighbors , modulo an additive constant . as we have done for pr, we define the _reduced -centrality _ as so , we can rewrite eq .( [ eq24 ] ) as where is again the degree of node .we can apply a similar argument as in section [ subsubsec3.3 ] .the sum over the neighbors of can be approximated as , where is the average of the reduced over the whole graph .the approximation is the more valid , the larger the number of summands . in this way , from eq .( [ eq25 ] ) we see that the reduced of a node is proportional to its degree , if the latter is large enough .this result is independent of the specific graph we consider , and we have verified it numerically for many types of networks . in fig .[ fig6_0 ] we show the distribution of reduced for different choices of the parameter for the sample of the web graph we analyzed in fig . [ fig5a ] .the curves closely follow the decay of the degree distribution .   reduced on undirected graphs .variability of reduced distribution with on the domain .gov of the world wide web .the degree distribution has a tail which follows fairly well a power law with exponent . to better show the agreement we have shifted the curves such that the tails overlap.,width=294 ]   the meaning of the eigenvalue equations ( [ eq7 ] ) and ( [ eq8 ] ) is quite simple .the hub score of a node is the sum of the hub scores of the in - neighbors of the out - neighbors of the node .the authority score of a node is the sum of the authority scores of the out - neighbors of the in - neighbors of the node ( fig .[ fig6a ] ) .the authority score of the node in the center is proportional to the sum of the authority scores of the out - neighbors ( blue squares ) of the in - neighbors ( red circles ) of the node.,width=264 ]  let us suppose that the nodes have the same outdegree .the authority score of a node is given by the sum of terms , where is the indegree of .in fact , node has in - neighbors , each of them having out - neighbors .if is large , the number of summands is very large , and can be approximated by the average value of the authority score over the whole graph , times .this approximation is the more valid , the larger and .we conclude that on a directed graph with constant outdegree the distribution of the authority scores will have the same tail as the indegree distribution .this is clearly illustrated in fig .[ fig7 ] . for the hubscores it is not possible to make predictions ; the sum that delivers the hub score of a node can not be approximated through other graph variables in most cases .the extension of the hits scores to the case of undirected graphs is not interesting . in this case , so and the hub and authority scores are identical .moreover , they coincide with ev , as the matrices and have the same eigenvectors .distribution of the authority scores versus indegree distribution .( left ) dms graph with nodes , and .( right ) dms graph with nodes , and ., width=302 ]in the previous sections we have investigated the distributions of spectral centrality measures and their similarities . as we have mentioned in the introduction , centrality measures are used to rank nodes . in this sectionwe shall compare the rankings obtained with different centrality measures . in order to compare two rankings we adopt kendall s  , a widely used index in this type of analysis .kendall s ranges from ( perfect correlation ) to ( perfect anticorrelation ) . in table[ tab1 ] we show the cross - comparisons between all centrality measures we discuss in this work , for a dms directed graph . for completeness we have included the outdegree as well .as we can see , pr , and the authority scores are well correlated with indegree and with each other , whereas the other coefficients are small or negative ; has a strong correlation with outdegree as well .  .kendalls for each pair of centrality measures computed for a dms directed graph , with , and .[ cols="^,>",options="header " , ]centrality measures are very important to understand the properties of the nodes of complex networks and their topological roles .we have studied the most important centrality measures based on properties of graph matrices : pagerank , eigenvector centrality , and the hub and authority scores of hits .all these measures deduce the importance of a node in a self - consistent way from the importance of its nearest neighbors and , in the case of the hits scores , of its next - to - nearest neighbors .we have resumed some recent results on pagerank distributions on particular types of tree - like graphs . on those graphs, the distribution of pagerank in the limit decays as a power law with exponent .the same is true for -centrality , because its defining equation is formally equivalent to the equation for pagerank in the limit .these results on centrality distributions are likely to be true for an extended class of graphs , where there is a flow from the outermost nodes ( leaves ) to a sink .we have also seen that , on any graph , in the limit , the reduced pagerank of a node , i.e. the contribution of the random walk process to the measure , is simply proportional to the indegree of the node , if the nodes have ( about ) the same outdegree .we have studied for the first time the extension of pagerank to the case of undirected networks , finding that the reduced pagerank of a node is proportional to its degree , for large degrees , for any graph and value of .we proposed a simple explanation of this effect based on the central limit theorem , and verified numerically in several cases that the argument holds .similarly , the reduced -centrality of a node is also proportional to its degree , for large degrees , on any graph . with the same type of argument it is possible to show that the authority score of a node is proportional to its indegree , for large indegrees , when the outdegrees of all nodes are ( approximately ) the same .we conclude that there are often strong relations between our centrality measured and ( in)degree : some relations hold on particular graphs and/or limits , others are more general .these findings imply that the measures are often strongly correlated with each other .we have indeed seen that the rankings of nodes according to the centrality measures we have considered are quite close to each other for indegree , pagerank , eigenvector centrality and authority score on graphs built with the prescription of dorogovtsev , mendes and samukhin .we have shown in the paper that these graphs have special properties , and that some measures may be correlated to each other . instead ,on real graphs , like the networks of political blogs and the sample of the web graph we have considered , the structure is less regular and the measures are far less correlated to each other , as confirmed by the small values of the kendall s for each pair of centrality measures .this means that , for practical purposes , and in spite of their similarities , spectral centrality measures look at nodes from different perspectives , and allow to diversify their roles within the network , obtaining in this way more information about the importance of nodes .the scores computed from spectral centrality measures can complement the information about node s centrality derived from more traditional measures like node betweenness  .this is especially important for directed graphs , where node betweenness , as well as other measures based on geodesic paths , like closeness  , are not well defined .

Abstract: complex networks are characterized by heterogeneous distributions of the degree of nodes , which produce a large diversification of the roles of the nodes within the network . several centrality measures have been introduced to rank nodes based on their topological importance within a graph . here we review and compare centrality measures based on spectral properties of graph matrices . we shall focus on pagerank , eigenvector centrality and the hub / authority scores of hits . we derive simple relations between the measures and the ( in)degree of the nodes , in some limits . we also compare the rankings obtained with different centrality measures .