as wireless devices become increasingly pervasive and essential , they are becoming both a target for attack and the very weapon with which such an attack can be carried out .traditional high - level computer and network security techniques can , and must , play an important role in combating such attacks , but the wireless environment presents both the means and the opportunity for new forms of intrusion .the devices that comprise a wireless network are low - cost commodity items that are easily available to potential intruders and also easily modifiable for such intrusion .in particular , wireless networks are open to intrusion from the outside without the need for a physical connection and , as a result , techniques that would provide a high level of security in a wired network have proven inadequate in a wireless network , as many motivated groups of students have readily demonstrated .  although conventional cryptographic security mechanisms are essential to securing wireless networks , these techniques do not directly leverage the unique properties of the wireless domain to address security threats .the physical properties of the wireless medium are a powerful source of domain - specific information that can be used to complement and enhance traditional security mechanisms . in this paper, we propose a cross - layer approach to augment the security of wireless networks for indoor wireless environments .in particular , we believe that the nature of the wireless medium can be turned to the advantage of the network engineer when trying to secure wireless communications .the enabling factor in our approach is that , in the rich multipath environment typical of wireless scenarios , the response of the medium along any transmit - receive path is _ frequency - selective _ ( or in the time domain , _ dispersive _ ) in a way that is _ location - specific_. this means :  1 .the channel can be specified by a number of complex samples either in the frequency domain ( a set of complex gains at a set of frequencies ) or the time domain ( a set of impulse response samples at a set of time delays ) .such sets of numbers decorrelate from one transmit - receive path to another if the paths are separated by the order of an rf wavelength or more .while using the physical layer to enhance security might seem to be a radical paradigm shift for wireless systems , we note that this is not the first time that multipath and advanced physical layer methods have proven advantageous .specifically , we are encouraged in our belief by two notable parallel paradigm shifts in wireless systems : ( 1 ) code division multiple access ( cdma ) systems , where the use of rake processing transforms multipath into a diversity - enhancing benefit ; and ( 2 ) multiple - input multiple - output ( mimo ) antenna techniques , which transform scatter - induced rayleigh fading into a capacity - enhancing benefit .note that there have been recent efforts in studying the information and secrecy capacity , that can be achieved by using the radio channel information .in contrast , this paper studies the feasibility of using such radio channel information .it does so by explicitly devising hypothesis testing procedures to estimate and track the radio channel for authentication purposes .we begin ( section [ sec : relatedworks ] ) by reviewing some related work .then ( section [ sec : overview ] ) , we provide an overview of our proposed phy - layer authentication service .we next present a general time - variant channel model ( section [ sec_channel ] ) that we will use as the basis for our discussions in this paper . in section [ sec_test ] ,we describe a hypothesis testing framework for physical layer authentication . in section [ sec_sim ], we present an overview of our simulation approach .we present our simulation results in section [ sec_results ] , and wrap up the paper in section [ sec_conclusion ] with concluding remarks .in commodity networks , such as 802.11 networks , it is easy for a device to alter its mac address and claim to be another device by simply issuing an ifconfig command .this weakness is a serious threat , and there are numerous attacks , ranging from session hijacking to attacks on access control lists , which are facilitated by the fact that an adversarial device may masquerade as another device . in response, researchers have proposed using physical layer information to enhance wireless security . for example , spectral analysis has been used to identify the type of wireless network interface card ( nic ) , and thus to discriminate among users with different nics . a similar method ,radio frequency fingerprinting , discriminates wireless devices according to the transient behavior of their transmitted signals . for more general networks ,the clock skew characteristic of devices has been viewed as a remote fingerprint of devices over the internet . in addition , the inherent variability in the construction of various digital devices has been used to detect intrusion .more recently , the wireless channel has been explored as a new form of fingerprint for wireless security .the reciprocity and rich multipath of the ultrawideband channel has been used as a means to establish encryption keys . in , a practical scheme to discriminate between transmitterswas proposed and identifies mobile devices by tracking measurements of signal strength from multiple access points .a similar approach was considered for sensor networks in .concurrent to these efforts , the present authors have built a significance test that exploits the spatial variability of propagation to enhance the authentication in the stationary , time - invariant channel . in this paper , we have significantly expanded the method to cover a more generalized channel , where there are time variations due to changes in the environment .as in , however , the ends of the link remain stationary , as might be the case for a population of users sitting in a room or airport terminal .we will see that , in some cases , the time variations _ improve _ the authentication .authentication is traditionally associated with the assurance that a communication comes from a specific entity . in the context of physical layer authentication , however , we are not interested in identity , _ per se _ , but rather are interested in recognizing a particular transmitting device .the ability to distinguish between different transmitters would not replace traditional identity - based authentication , but would be particularly valuable as a wireless system enhancement .such an approach would be beneficial for scenarios where managing cryptographic key material is difficult , and further would reduce the load placed on higher - layer authentication buffers .  here, we borrow from the conventional terminology of the security community by introducing three different parties : alice , bob and eve . for our purposes, these three entities may be thought of as wireless transmitters / receivers that are potentially located in spatially separated positions , as depicted in fig .[ fig : seville ] .our two legal " protagonists are the usual alice and bob , and for the sake of discussion throughout this paper , alice will serve as the transmitter that initiates communication , while bob will serve as the intended receiver . their nefarious adversary , eve , will serve as an active adversary that injects undesirable communications into the medium in the hopes of spoofing alice .our security objective , broadly speaking , is to provide authentication between alice and bob , despite the presence of eve .since eve is within range of alice and bob , and capable of injecting her own signals into the environment to impersonate alice , bob must have the ability to differentiate between legitimate signals from alice and illegitimate signals from eve .consider a simple transmitter identification protocol in which bob seeks to verify that alice is the transmitter .suppose that alice transmits probes into the channel at a rate sufficient to assure temporal coherence between channel estimates and that , prior to eve s arrival , bob has estimated the alice - bob channel .eve wishes to convince bob that she is alice .bob will require that each information - carrying transmission be accompanied by an authenticator signal .the channel response to a transmitted signal between alice and bob is a result of the multipath environment .bob may use the received version of the authenticator signal to estimate the channel response and compare this with a previous record for the alice - bob channel .if the two channel estimates are close " to each other , then bob will conclude that the source of the message is the same as the source of the previously sent message .if the channel estimates are not similar , then bob should conclude that the source is likely not alice .there are several important issues related to such a procedure that should be addressed before it can be a viable authentication mechanism .first is the specification of the authenticator signal that is used to probe the channel .there are many standardized techniques to probe the channel , ranging from pulse - style probing ( including pn sequences ) to multi - tonal probing , and we may use these techniques to estimate the channel response .regardless of what probing method is employed , the channel response can be characterized in the frequency domain , and throughout this paper we will represent our channels in that domain .a second issue is that in a richly scattered multipath environment ( typical of indoor wireless environments ) , it is difficult for an adversary to create or precisely model a waveform that is transmitted and received by entities that are more than a wavelength away from the adversary .this assertion is supported by the well - known jakes uniform scattering model , which states that the received signal rapidly decorrelates over a distance of roughly half a wavelength , and that spatial separation of one to two wavelengths is sufficient for assuming independent fading paths .the implication of such a scattering model in a transmitter identification application remains to be tested , and a key objective of this study is to examine the utility of a typical indoor multipath environment for discriminating between alice - bob and eve - bob channels .  finally , it should also be noted that the channel response may change with time due to changes in the environment ( people moving , doors opening or closing ) and in practice it will be necessary to guarantee the continuity of the authentication procedure by probing the channel at time intervals less than the channel s coherence time " .this paper examines the ability to authenticate transmitters in such a time - variant environment , and serves to illustrate the potential for new forms of physical layer security .we assume that bob first measures and stores the frequency response of the channel connecting alice with him . due to his receiver thermal noise , bob stores a noisy version of the channel response , .after awhile , he has to decide whether a transmitting terminal is still alice , based on a noisy measured version , , of that terminal s channel response to bob . by sampling and at  ] is the sample from at the -th tone at a sampling time of .more specifically , =x(kt;f_o - w/2+m \delta f) ] is the zero - mean variable part at time , and  ] is independent of  ] over time .we set , since the receiver can not resolve two components with time difference smaller than the inverse of the bandwidth .the frequency response of the variable part is the fourier transform of in terms of , &={\cal f } \{h(t;\tau)\}|_{t = kt , f = f_o - w/2+m \delta f}\nonumber\\&=\sum_{l=0}^{\infty } a_l[k]e^{-j2\pi ( f_o - w/2+m \delta f)l \delta } , \label{e_epsil}\end{aligned}\ ] ] where =a_l(kt) ] , i.e. , =\textrm{var}[a_l[k]]=\sigma _ t^2(1-e^{-\gamma\delta\tau})e^{- \gamma \delta \tau l } , \label{e_var_al}\end{aligned}\ ] ] where is the inverse of the average delay spread , is the coherence bandwidth of the variable part , and is the average power of  ] , i.e. , =a a_l[k-1]+\sqrt{(1-a^2)p_\tau[l]}u_l[k ] , \label{e_at}\end{aligned}\ ] ] where the ar coefficient denotes the similarity of two values spaced by and the random component \sim cn(0,1) ] .as pointed out in section [ sec_intro ] , in a typically rich scattering environment , the radio channel response decorrelates quite rapidly in space .later , we will cite the use of the ray - tracing software wise to emulate the spatial correlation characteristics of the _ fixed _ part of the channel response ( ) .as to the _ variable _ part , however , we consider the two extreme cases :  \1 ) _ spatially independent and identically distributed _the frequency response sample of the channel between eve and bob can be written as =\overline{h}_{e , m}+\epsilon_{e , m}[k]+n_{e , m}[k ] , \label{eq_che1}\end{aligned}\ ] ] where ,  ] ; and  ] are independent identically distributed ( i.i.d . ) .\2 ) _ complete spatially correlated variation _( =\epsilon_{a , m}[k] ] , * r * and are the covariance matrix of -{\textbf h}_{a}[k-1] ] , where the elements are i.i.d . , and , .thus the test statistic is a chi - square random variable with degrees of freedom , i.e. , .we define the rejection region for as .thus , the false alarm rate " ( or type i error ) is ; and the miss rate " ( or type ii error ) is given by ( [ e_beta_0 ] ) ,  -\textbf{h } _ { a}[k-1])^h \textbf{r}^{-1}({\textbf h}_{e , t}[k]-\textbf{h } _ { a}[k-1 ] ) < { \textrm f}^{-1}_{\chi^2_{2m}}(1-\alpha)\},\label{e_beta_0}\ ] ]  where is the cdf of the random variable and is the inverse function of .for a specified , the threshold of the test is , and the miss rate can be obtained by numerical methods .   when the variation is independent over tones ( i.e. , ) , the covariance matrices of eq .( [ e_r1 ] ) and ( [ e_g1 ] ) become -\textbf{h } _ { a}[k-1]]=(2(1-a ) \sigma _ t ^2 + 2 \sigma _n ^2)\textbf{i}\nonumber\\ \textbf{g}&=\textrm{cov}[\textbf{h}_{e}[k]-\textbf{h } _ { a}[k-1]]=(2\sigma _ t ^2 + 2 \sigma _n ^2)\textbf{i},\end{aligned}\ ] ] where * i * is the identity matrix .thus the test statistic eq .( [ e_test_sta_1 ] ) becomes -\textbf{h}_{a}[k-1]|^2}{(1-a ) \sigma _ t ^2 + \sigma _n ^2}=z_2/\rho , \end{aligned}\ ] ] where   it is easy to see that , under , the test statistic is a non - central chi - square distribution with order , i.e. , with non - central parameter thus , the miss rate for specified , ( [ e_beta_0 ] ) , can be written as    when the variation is totally correlated over tones ( i.e. , ) , the covariance matrices of ( [ e_r1 ] ) and ( [ e_g1 ] ) degrade to where * 1 * is a matrix with each element equals to 1 . again , we can use eq .( [ e_beta_0 ] ) to numerically calculate the miss rate for specified false alarm rate .when bob does not know the parameters , and , it is reasonable for him to use as the test statistic -\textbf{h } _ { a}[k-1]|^2.\end{aligned}\ ] ] in this case , we can obtain numerical results for the false alarm rate and miss rate for specified threshold , plotting vs. with as an implicit parameter .now we consider the other extreme case of spatial correlation , namely , =\epsilon_{a , m}[k] ] .thus , the test statistic under is non - central chi - square distributed , -{\textbf h}_{a}[k-1])|^2 \sim \chi^2_{2m,\mu} ] ( positive impact on performance ) .  when is negligible , the channel can be viewed approximately as a time - invariant one , wherein the miss rate is given by ( [ e_beta_2 ] ) . as rises ,the miss rate falls since the positive impact dominates .if the variation continues to rise and becomes very large , the miss rate begins to rise , as the need to raise the threshold helps eve and counteracts the positive impact .when becomes so large that both the fixed part of the channel response and the thermal noise are relatively negligible ( i.e. , , ) , then using ( [ e_beta_1 ] ) we can rewrite the miss rate as which is a function of the time - correlation of the temporal variation parameter ( ) , frequency sample size ( ) , and the false alarm rate ( ) . if the variation is strongly correlated in time ( ) , the miss rate can be less than that for the noise - dominated case , ( [ e_beta_2 ] ) , where the thermal noise is usually not negligible due to the limited transmit power .an illustration of this trend will be given later .  finally , we consider the impact of the spatial correlation of time variations .the miss rate with total spatial correlation , ( [ e_beta_end ] ) , decreases with in a manner that is proportional to the inverse of * r * , ( [ e_r1 ] ) , and thus rises with . since a strong spatial correlation of the time variation damages the spatial variability character of the channel , which is the basis of our scheme , it will degrade the system performance .in order to test the proposed scheme , it is necessary to model not only typical " channel responses , but the spatial variability of these responses . only in this waycan we discern the success in detecting would - be intruders like eve . to that end, we make use of the wise tool , a ray - tracing software package developed by bell laboratories .one input to wise is the 3-dimensional plan of a specific building , including walls , floors , ceilings and their material properties . with this information, wise can predict the rays at any receiver from any transmitter , including their amplitudes , phases and delays . from this , it is straightforward to construct the transmit - receive frequency response over any specified frequency interval ( bandwidth ) .we have done this for one particular office building ( the alcatel - lucent crawford hill laboratory in holmdel , nj ) , for which a top view of the first floor is shown in fig . [ fig_building ] .this floor of this building is 120 meters long , 14 meters wide and 4 meters high . for our numerical experiment, we placed bob in the hallway ( the filled - in circle ) at a height of 3 m. for the positions of alice and eve , we considered four rooms at the extremities of the building ( shown shaded ) .for each room , we assumed alice and eve both transmitted from a height of 2 m , each of them being anywhere on a uniform horizontal grid of points with 0.2-meter separations . with grid points in a room, there were possible pairs of alice - eve positions .for rooms 1 , 2 , 3 and 4 , the numbers of grid points were , 713 , 315 and 348 , respectively .for each alice - eve pair , ( 1 ) wise was used to generate the alice - bob and eve - bob average channel responses ( and ) ; and ( 2 ) the hypothesis test described above was used to compute for a specified .the set of all -values in a room were used to compute a room - specific mean , , for each of several selected combinations of bandwidth ( ) , number of frequency - domain samples ( ) , transmit power ( ) , and channel variation models .assume that , in conjunction with wise , we obtain the various transfer functions as dimensionless ratios ( e.g. , received _e_-field / transmitted _ e_-field ) .then the proper treatment of the noise variance , , in the hypothesis test is to define it as the receiver noise power per tone , , divided by the transmit power per tone , , where is the total transmit power . noting that , where is the thermal noise density in mw / hz , is the receiver noise figure , and is the measurement noise bandwidth per tone in hz , we can write where is in mw , and . we will henceforth refer to by its decibel value .let denote the ratio between and the value of averaged over the frequency samples ( or tones " ) and the receiver locations .we can thus write the standard deviation of the time variation as where can be regarded as a room parameter , and represents the relative magnitude of the time variation in a given room .in our simulations , we set ghz , ( 10 db noise figure ) , mw / hz , mhz , , and , unless specified otherwise ,  .as noted earlier , we place alice and eve on dense grids in each of four rooms at the corners of a particular building , with bob in the hallway , fig . [ fig_building ] .we obtained a miss rate for each alice - eve pair in each room , and then calculated the average mean value for each room in the building . among them , room # 4 , as the farthest room from bob , is likely to have the poorest performance in rejecting eve . in that sense , it lower - bounds the capabilities of our phy - layer authentication algorithm . for reasons of space , we will only present results for room # 4 , keeping in mind that they are essentially worst - case or close to it .figure [ fig_result_time_variance ] confirms the efficacy of the algorithm in the presence of channel time variations .we assume realistic system parameter values ( mw 1 w , and mhz ) , and find that most average miss rates are smaller than 0.01 .the per tone signal - to - noise ratio ( snr ) in the channel measurements ranges from -12.8 db to 14.2 db , with a media value of 6.4 db , if using mw , and mhz . also , as pointed out in section [ sec_test ] , our proposed algorithm can exploit the time variations to improve performance .for example , the miss rate falls from around 0.01 to when rises from to , with mw .the trend of these curves with time variation confirms the discussion in section [ subsec : variation ] , e.g. , the minimum average miss rate is a tradeoff between the positive impact of the time variation and its negative impact resulting from the rise of the threshold .moreover , the miss rate falls with the transmit power , as expected , since it reduces the measurement noise at the receiver .figure [ fig_result_w ] demonstrates the impact of the bandwidth and the coherence bandwidth .we note that the results for and are the lower and upper bounds , respectively , of the miss rates , as well as the asymptotic results for the high- and low - bandwidth regions .it is clear that frequency correlations degrade performance .a related finding is that the miss rate decreases with increasing bandwidth , , since the frequency response samples are more independent with larger .figure [ fig_result_m ] indicates that there is little benefit ( or even a deficit ) in increasing beyond , unless the frequency correlation is very small ( e.g. , ) with high transmit power . actually , the optimal sample size in terms of miss rate for specified measurement bandwidth decreases with the coherence bandwidth , because the noise power ( [ e_sigma_n ] ) rises with and the frequency - response samples are more correlated with larger .we see in fig .[ fig_result_unknown ] that the algorithm works even when bob does not know the key channel parameters , although it requires either more transmit power or greater tolerance for type ii errors .interestingly , the time - variation may still help here , e.g. , the miss rate falls as rises from 0.1 to 1 .how to set the test threshold in this case is an open topic .  finally , fig .[ fig_result_spatial_correlation ] shows that the system is very sensitive to the time - variation in an extreme case of full spatial correlation .it requires much more transmit power ( ) to reach the same miss rate performance .the reason is quite simple : the mechanism of our scheme is to utilize the spatial variability of the channel responses .the spatial correlation of the time variation decreases the overall spatial variability and thus degrades the performance .the above assertions apply as well to the other shaded rooms in fig . [ fig_building ] and , we can safely assume , to the other rooms in the building .we have described and studied a physical layer technique for enhancing authentication in a time - variant wireless environment .specifically , we assume the user terminals are stationary but changes in the environment produce additive time - varying changes in the channel responses .the technique uses channel frequency response measurements and hypothesis testing to discriminate between a legitimate user ( alice ) and a would - be intruder ( eve ) . with the ability to utilize the temporal - variation , it works even when the receiver does not know the key channel variation parameters , namely , the ar temporal coefficient , the coherence bandwidth and the standard deviation of the variation , although these parameters help reduce the miss rate if known .the algorithm has been verified in a typical in - building environments , where we used the ray - tracing tool wise to generate realistic average channel responses and used a multipath tapped delay line channel model for the temporal variation part of the channel response .simulation results have confirmed the efficacy of the algorithm for realistic values of the measurement bandwidth ( e.g. , mhz ) , number of response samples ( e.g. , ) and transmit power ( e.g. , mw ) .the miss rate is generally smaller than 0.01 , for a specified false alarm rate of 0.01 , in the presence of moderate channel time variations .we have found that the channel time variations can improve the performance , e.g. , the miss rate falls from around 0.01 to when the variation index rises from to , with mw .in addition , the miss rate decreases with the transmit power of the probing signal and the measurement bandwidth , and usually requires frequency samples of fewer than 10 .we have also shown that the time correlation of the channel variation is helpful , while coherence in the frequency and spatial domains are harmful .research is currently in progress to address the case of user terminal mobility .effort is also needed , for the stationary case , to explore the parameter space ( e.g. , the temporal coherence term ) ; devise means of setting the test threshold ; consider other buildings ; and conduct experiments to more accurately characterize the time - variation properties of indoor channels .moreover , we are working to integrate physical layer authentication into a holistic cross - layer framework for wireless security that will augment traditional higher - layer " network security mechanisms with physical layer methods .since \sim cn(0,p_\tau [ l]) ] and  ] has zero mean and is a gaussian random variable , since it is the linear combination of gaussian random variables .for the case with spatially independent time variation where  ] are independent identically distributed , from ( [ eq_cha ] ) and ( [ eq_che1 ] ) we have -h _ { a , m}[k-1]]\nonumber\\&= \textrm{var}[\epsilon _ { e , m}[k]-\epsilon _ { a , m}[k-1]+n_{e , m}[k]-n_{a , m}[k-1]]\nonumber\\ & = \textrm{var}[\epsilon _ { e , m}[k]]+\textrm{var}[\epsilon _ { a , m}[k-1]]\nonumber\\&\quad + \textrm{var}[n_{a , m}[k]]+\textrm{var}[n_{e , m}[k]]\nonumber\\&=2\sigma^2_t+2\sigma_n^2\end{aligned}\ ] ] and , -h _ { a , m}[k-1],h _ { e , n}[k]-h _ { a , n}[k-1]]\nonumber\\&= \textrm{cov}[\epsilon _ { e , m}[k]-\epsilon _ { a , m}[k-1],\epsilon _ { e , n}[k]-\epsilon _ { a , n}[k-1]]\nonumber\\&=e[\epsilon _ { e , m}[k]\epsilon ^*_{e , n}[k]]+e[\epsilon _ { a , m}[k-1]\epsilon ^*_{a , n}[k-1]]\nonumber\\&=\frac{2\sigma^2_t(1-e^{-\gamma \delta \tau})}{1-e^{-\gamma \delta \tau+j2\pi ( n - m)\delta \tau \delta f}}=r(m - n)/(1-a)\end{aligned}\ ] ] from ( [ eq_cha ] ) and ( [ eq_che1 ] ) we also see that -\textbf{h } _ { a}[k-1 ] ] = \overline{\textbf{h}}_e-\overline{\textbf{h}}_a$ ] . the other part is similar to that of relationship 1 .n. borisov , i. goldberg , and d. wagner , `` intercepting mobile communications : the insecurity of 802.11 , '' in proc .acm annual international conference on mobile computing and networking ( mobicom ) , pp .180189 , sept .2002 .t. daniels , m. mina , and s. f. russell , `` short paper : a signal fingerprinting paradigm for general physical layer and sensor network security and assurance , '' in _ proc .ieee / create net secure commum ._ , pp . 219 221 , sept . 2005 .m. demirbas and y. song , `` an rssi - based scheme for sybil attack detection in wireless sensor networks , '' in _ proc .international workshop on advanced experimental activity _ , pp .564 570 , june , 2006 .l. xiao , l. greenstein , n. mandayam , and w. trappe , `` fingerprints in the ether : using the physical layer for wireless authentication , '' in _ proc .ieee international conference on communications _ , glasgow , scotland , june 2007 .v. erceg , d. g. michelson , s. s. ghassemzadeh , l. j. greenstein , a. j. rustako , p. b. guerlain , m. k. dennison , r. s. roman , d. j. barnickel , s. c. wang , and r.r .miller , `` a model for the multipath delay profile of fixed wireless channels , '' , vol .17 , pp . 399410 , 1999 .bello and b.d .nelin , `` the effect of frequency selective fading on the binary error probability of incoherent and differentially coherent matched filter receivers , '' , vol .cs-11 , pp .170186 , june 1963 .s. j. fortune , d. h. gay , b. w. kernighan , o. landron , m. h. wright , and r. a. valenzuela , `` wise design of indoor wireless systems : practical computation and optimization , '' , vol .58 - 68 , mar .

Abstract: the wireless medium contains domain - specific information that can be used to complement and enhance traditional security mechanisms . in this paper we propose ways to exploit the spatial variability of the radio channel response in a rich scattering environment , as is typical of indoor environments . specifically , we describe a physical - layer authentication algorithm that utilizes channel probing and hypothesis testing to determine whether current and prior communication attempts are made by the same transmit terminal . in this way , legitimate users can be reliably authenticated and false users can be reliably detected . we analyze the ability of a receiver to discriminate between transmitters ( users ) according to their channel frequency responses . this work is based on a generalized channel response with both spatial and temporal variability , and considers correlations among the time , frequency and spatial domains . simulation results , using the ray - tracing tool wise to generate the time - averaged response , verify the efficacy of the approach under realistic channel conditions , as well as its capability to work under unknown channel variations .


acquiring knowledge about a sampled space from a point data has become a key problem in many areas of science and engineering .the sampled space could be a hidden manifold sitting in some high dimensions , or could be a compact subset of some euclidean space .topological information such as the rank of the homology groups , or their persistent behavior can divulge important features of the hidden space .therefore , a considerable effort has ensued to extract topological information from point data in recent years  . with the advent of advanced technologies ,the data is often generated in abundance .mixed with the burden of high dimensionality , large data sets pose a challenge to the processing resource . as a result ,some recent investigations have focused on how to use a lighter data structure or sparsify the input which aids a faster computation , but still guarantees that the output inference is correct .point data by themselves do not have interesting topology .so , a foremost step in topology inference is to impose a structure such as a simplicial complex onto it .the delaunay , ech , vietoris - rips , and witness complexes are some of the most commonly proposed complexes for this purpose . among these ,vietoris - rips ( rips in short ) and witness complexes  have been favored because they can be constructed with simple computations .rips complexes are easy to construct as they can be built from a graph by recognizing the cliques in it .however , the presence of simplices corresponding to all cliques makes its size quite large . even in three dimensions with a few thousand points ,the size of the rips complex can be an obstacle , if not a stopper , for further processing .witness complexes , on the other hand , have too few simplices to capture the topology of the sampled space in dimensions three or more  . to tackle this issue , boissonnat et al . suggested modifications to the original definition of witness complex  .this enlarges the witness complexes but makes it more complicated and costly to compute .we investigate a new complex , a version of which was originally introduced in  for the application of sensor network routing .we set up a more general definition and call it the _ graph induced complex_. we provide new theoretical understanding of the graph induced complex in terms of topology inference . in particular , we show that , when equipped with appropriate metric , this complex can decipher the topology from data .it retains the simplicity of the rips complex as well as the sparsity of the witness complex .its construction resembles the sparsified rips complex proposed in  and also the combinatorial delaunay triangulation proposed in  , but it does not build a rips complex on the subsample and thus is sparser than the rips complex with the same set of vertices .this fact makes a real difference in practice as our preliminary experiments show .the idea of graph induced complex also bears similarity to the geodesic delaunay triangulation which was proposed to recover the topology of a bounded planar region ( with holes ) from point samples .our work extends it , as well as investigates its theoretical properties , to more general setting beyond the planar case .given a graph on a point data equipped with a metric , one can build a graph induced complex on a subsample by throwing in a simplex with a vertex set if a set of points in , each being closest to exactly one vertex in , forms a clique .figure [ fig : gic ] shows a graph induced complex for a point data in the plane .subsampled points are the darker vertices .input points are grouped according to the proximity to the subsampled vertices ( indicated with a voronoi partition ) .the shaded triangle enlarged on right is in the graph induced complex since there is a -clique in the input graph whose 3 vertices have 3 different closest point in the subsample .observe that , in this example , the graph induced complex has the same homology as the sampled space . figure [ fig : comparison ] shows experimental results on two data sets , 40,000 sample points from a klein bottle in and 15,000 sample points from the primary circle of natural image data considered in  .the graphs connecting any two points within unit distance for klein bottle and unit distance for the primary circle were taken as input for the graph induced complexes .the -skeleton of the rips complexes for these parameters have and simplices respectively .these sizes are too large to carry out fast computations .  [cols="^,^ " , ]in this section we consider domains beyond surfaces in .let be a -manifold embedded in and a discrete sample of .we observe that the overall setup in section [ surface - topoinfer ] for inferring -homology from the graph induced complex generalizes easily to higher dimensions . the inclusion map still induces an isomorphism at the homology level if and are chosen appropriately . in that case, the map remains injective by the same argument as before .the main trouble arises when we try to prove that it is also surjective .observe that , to prove the surjectivity of , we used the fact that the restricted voronoi cell in is connected ( proposition [ rips - connect ] ) . unfortunately , this is not necessarily true in high dimensions given the counterexamples in  . to overcome this impediment we change the distance function replacing the euclidean distance with the graph distance while building .specifically , we still consider to be the graph connecting points in with euclidean distance or less , but take to be a -sparse -sample of where the graph distance is defined with the euclidean lengths as the edge weights .then , we consider .  as before ,let be the set of points nearest to a point with respect to .the modification in distance function immediately allows us to claim that the -skeleton of is connected , which was needed to claim that is surjective . is connected , and thus is surjective .[ rpq - connected ]  let be an -sample of an embedded smooth and compact manifold with reach , and a -sample of . for ,the map is an isomorphism where is the simplicial map induced by the nearest point map .[ cor : manifoldgeodesic ]   in this subsection we show that the subsample can be made leaner .the main insight is that we can define a feature size larger than the reach which permits us to subsample more sparsely with respect to this larger feature size .gao et al . considered a similar feature size for the same reason of requiring sparser sampling for a two dimensional shape . herewe show that such a sparser sample is also adequate for determining of manifolds in high dimensions .our experimental results in figure [ fig : comparison ] suggest that one can obtain information about from a very sparse sample in practice .let be a simplicial complex with non - negative weights on its edges .we define _ homological loop feature size _ as this feature size is very similar to the _ systolic feature size _ of a compact metric space  which is the length of the shortest non - contractible loop in . our definition of when applied to a metric space becomes larger than or equal to .notice that every loop of with length less than is null - homologous in .let be a -sample of as before but with .let be the simplicial map as defined earlier .we aim to show that the induced homomorphism on the first homology is injective .since we use graph distances , proposition [ rpq - connected ] remains valid and hence remains to be surjective .however , we can not claim to be an isomorphism because could be larger than required .thus , we can not use to infer to be injective as before .nevertheless , we can prove the following result using a different approach .[ injectivity theorem ] if is a -sample of for , then is an isomorphism .[ thm : injectivityforsys ]  we only need to show that is injective , as its surjectivity follows from proposition [ rpq - connected ] .to show the injectivity , it suffices to show that has a trivial kernel .let be any triangle in .if under the chain map , every cycle in the preimage of the boundary cycle is null homologous , then every null homologous cycle in has only null homologous cycles in its preimage .this is true due to the fact that a bounded cycle is a sum of boundaries of triangles , and that the chain map is surjective ( see the proof of proposition [ surjectivity ] ) .below we show that under the chain map , every cycle in the preimage of the boundary cycle of any triangle indeed is null homologous .it would then follow that the kernel of is trivial .let be any cycle in the preimage of .we have where be any edge of of .let be any maximal subpath of so that ( figure [ hlfs - fig](a ) ) .for each such , we construct a cycle so that is null homologous and is homologous to . therefore , our problem of showing is null homologous reduces to the problem of showing every is null - homologous .we construct as follows . by the construction of ,there is a triangle such that with , .consider the shortest paths and in from to and from to respectively .observe that all vertices in and are mapped to and respectively by since we are using the graph - induced distance to construct .take to be the chain ; refer to figure [ hlfs - fig](a ) . with this choice, we have and hence is homologous to as promised . to prove null - homologous, we construct a homologous path to which gives a homologous cycle to .we then prove that is null - homologous .call an edge in _ bicolored _ if its two end - points are mapped to two distinct vertices by ; otherwise , is _ unicolored_. a -chain from is _ unicolored _ if it has only unicolored edges . for simplicity, we assume that vertices from are all contained in because is always homologous to a path containing vertices mapped only to or . in this case , can be decomposed into a set of bicolored edges together with a set of unicolored chains such that .in particular , can be written as see figure [ hlfs - fig](b ) for an illustration .let be a unicolored chain with two boundary points so that for any simplex , . then , is homologous to the chain .[ path - reduce ]  given the chain with subchains as in equation [ eqn : gamma ] , we convert it to a homologous chain replace with in to obtain a homologous cycle ; refer to figure [ hlfs - fig ] . observe that is the sum of cycles ( diamonds ) that have two unicolored chains and two bicolored edges as shown in figure [ hlfs - fig](d ) .such a cycle has length at most .this is because each unicolored chain in has at most two shortest paths of the form and ( or and ) that have lengths or less ( is a -sample of ) , and the two bicolored edges have lengths at most in .the cycle is null homologous because its length is it follows that is null homologous as we are required to prove .we only need to show claim [ path - reduce ] to finish the proof .let be the sequence of vertices on the unicolored chain ( path ) .consider the shortest paths for each on this path .the length of the cycle is at most for each $ ] .therefore , it is null homologous by our assumption .we have . therefore , and are homologous .so far we have focused on -homology . in this sectionwe extend the domain to compact subspaces of euclidean spaces and consider homology groups of all dimensions .this generality comes at the expense of additional computations . unlike previous approaches that allow us to infer the -homology of the sampled manifold by computing directly the same for the graph induced complexes ,now we need to compute the persistent homology  induced by simplicial maps .the well - known algorithms for computing persistent homology  work for maps induced by inclusions .in a contemporary paper  , we present an algorithm that can compute the persistent homology induced by simplicial maps .let be a compact set and be its offset with . since it is difficult to compute from a sample  , we aim for computing the homology groups for the offset .let denote the the _ weak feature size _ which is defined as the smallest positive critical value of the distance function to  .we prove that the persistent homology of the graph induced complex defined with either euclidean or graph distance provides the correct homology of where .specifically , the image of induced by a simplicial map becomes isomorphic to for appropriate and .we recall the following result from .[ lemma : six - seq ] if the sequence of homomorphisms between finite dimensional vector spaces satisfies that , then .  let and be subsamples of where is a -sparse -sample and is a -sparse -sample for .consider the interleaving sequence between the graph induced and rips complexes , ^ -{i_1 } \ar@{->}[rd]^-{h_1 }  & \  { { \cal r}}^{\alpha+2\delta}(p ) \\ar@{^{(}->}[r]^-{i_2 } & \ { { \cal r}}^{4(\alpha+2\delta)}(p ) \\ar@{^{(}->}[r]^-{i_3 } \ar@{->}[d]^-{h_2 }  & \ { { \cal r}}^{4(\alpha+2\delta)+2\delta^{'}}(p )  \\ & { \mathcal{g}}^{\alpha}(p , q , d )  \ar@{.>}[r]^-{h } \ar@{^{(}->}[u]^-{j_1 } & { \mathcal{g}}^{4(\alpha+2\delta)}(p , q',d ) \  \ar@{^{(}->}[ru]^-{j_2 }  & } \ ] ] where and are inclusions and . by proposition [ prop : contg ] , and are simplicial maps . therefore , is also a simplicial map as composition of simplicial maps . in particular, is the simplicial map induced by the vertex map that maps each point to its closest point in .we prove that where and , and fall in appropriate ranges .let be a compact space .let and be an -sample of .let be a -sparse -sample of and be a -sparse -sample of where is either euclidean or graph distance and .+ if and , then  where is induced by in diagram [ diag : gic - rips ] .the diagram [ diag : gic - rips ] is not commutative in general .however , it is commutative at the homology level .proposition [ prop : contg ] makes the two triangles at the left and right commutative .the middle square commutes by definition of . now consider the sequence , ^ -{{h_1 } _ { * } }  & { { \sf h}}_{k}({\mathcal{g}}^{\alpha}(p , q , d ) )  \ar@{->}[r]^-{{j_1 } _ { * } } & \  { { \sf h}}_{k}({{\cal r}}^{\alpha+2\delta}(p ) )  \\ \  \ar@{->}[r]^-{{i_2 } _ { * } }  & \ { { \sf h}}_{k}({{\cal r}}^{4(\alpha+2\delta)}(p ) ) \  \ar@{->}[r]^-{{h_2 } _ { * } }  & { { \sf h}}_{k}({\mathcal{g}}^{4(\alpha+2\delta)}(p , q',d ) ) \  \ar@{->}[r]^-{{j_2 } _ { * } }  & \ { { \sf h}}_{k}({{\cal r}}^{4(\alpha+2\delta)+2\delta^{'}}(p ) )  } \label{seq}\end{aligned}\ ] ] consider the sequence of inclusions at the upper level of the diagram ( 1 ) . since and , we have that by theorem of . considering the diagram in ( 1 ) and the sequence in ( [ seq ] ) we have letting , the rightmost isomorphism in ( [ seq3 ] ) allows us to claim that by applying proposition [ lemma : six - seq ] to the sequence ( [ seq ] )in this work , we investigated the graph induced complex that can be built upon a given point cloud data and a suitable graph connecting them .this complex , to some extent , has the advantages of both rips and witness complexes .we have identified several of its topological properties that can evidently be useful in extracting information from point data even in high dimensions .  in section [ compact - sec ] , we have shown how one can infer the homology groups of a compact set using the persistent homology of a pair of graph induced complexes constructed with two values of .one can consider a filtration of with being sparsified for increasing values of and .then , one can obtain a persistence diagram  out of this ` full filtration ' using our recently proposed algorithm for computing the topological persistence for filtrations connected with simplicial maps  .the algorithm will collapse vertices progressing through the filtration and hence will keep the size of the complex in question contained . relating this persistence diagram to that of a filtration obtained by a related rips filtrationis an interesting question . in a subsequent work, we have addressed this question in  .finding other applications where the graph induced complex becomes useful also remains open for further investigations .we acknowledge the support of the nsf grants ccf 1116258 and ccf-1064416 and thank all the referees for their valuable comments .  99  h. adams and g. carlsson . on the nonlinear statistics of range image patches ._ siam j. img .* 2 * ( 2009 ) , 110117 .d. attali , a. lieutier , and d. salinas .vietoris - rips complexes also provide topologically correct reconstruction of sampled shapes .27th annu .( 2011 ) , 491500 .n. amenta , s. choi , t. k. dey and n. leekha . a simple algorithm for homeomorphic surface reconstruction . _ int .j. comput .* 12 * ( 2002 ) , 125141 .boissonnat , l. j. guibas , and s. y. oudot .manifold reconstruction in arbitrary dimensions using witness complexes ._ discrete comput .* 42 * ( 2009 ) , 37 - 70 .boissonnat and c. maria .the simplex tree : an efficient data structure for general simplicial complexes .esa , lecture notes in computer science _ , vol .7501 ( 2012 ) , 731742 .g. carlsson and v. de silva .topological approximation by small simplicial complexes , 2003 .preprint .f. chazal , l. j. guibas , s. y. oudot , p. skraba .analysis of scalar fields over point cloud data ._ discrete comput .geom . _ * 46 * ( 2011 ) , 743775 .f. chazal and a. lieutier . the -medial axis .models _ * 67 * ( 2005 ) , 304331 .  f. chazal and s. oudot . towards persistence - based reconstruction in euclidean spaces .( 2008 ) , 232241 .cheng , t. k. dey , and e. a. ramos .manifold reconstruction from point samples .16th sympos . discrete algorithms _ ( 2005 ) , 10181027 .d. cohen - steiner , h. edelsbrunner , and j. l. harer .stability of persistence diagrams ._ discrete comput .* 37 * ( 2007 ) , 103 - 120 .  t. k. dey . curve and surface reconstruction : algorithms with mathematical analysis . cambridge university press , new york , 2007 .  t. k. dey , f. fan , and y. wang .computing topological persistence for simplicial maps .arxiv:1208.5018v3 [ cs.cg ] , 2013 .t. k. dey and y. wang .reeb graphs : approximation and persistence .27th annu .( 2011 ) , 226235 .extended version available from authors web - pages .h. edelsbrunner and j. harer .computational topology .american mathematical society , 2009 .q. fang , j. gao , l. guibas , v. de silva , and l. zhang .glider : gradient landmark - based distributed routing for sensor networks .infocom 2005 _ ( 2005 ) , 339 - 350 .j. gao , l. guibas , s. oudot , and y. wang .geodesic delaunay triangulation and witness complex in the plane . _algorithms ( talg ) _ * 6 * ( 2010 ) , 67:167:47 .  r. ghrist .barcodes : the persistent topology of data .* 45 * ( 2008 ) , 61 - 75 .t. gonzalez .clustering to minimize the maximum inter - cluster distance .theoretical computer science , * 38 * ( 1985 ) , 293 - 306 .on the vietoris - rips complexes and a cohomology theory for metric spaces ._ prospects in topology : proc .( 1994 ) , 175188 .james r. munkres .elements of algebraic topology . addison wesley publishing company , menlo park , 1984 .d. sheehy .linear - size approximations to the vietoris - rips filtration .( 2012 ) , 239247 .v. de silva and g. carlsson .topological estimation using witness complexessympos . point based graphics _ ( 2004 ) , 157166 .first , we present an elementary geometric result that we need to use in the proof . let denote the cone with apex , axis in the direction of and aperture .  given a ball with radius and center , let be an arbitrary point on the surface of sphere .consider the two nested cones and with the same axis .let be any point from the intersection of the ball and the inner cone ; that is , .let be an arbitrary point from the boundary of outside the outer cone ; that is , , and .then we have that .[ claim : twocones ]  denote and . because is outside of the outer - cone with aperture , and is inside of the inner cone of aperture , we have that we have .now consider the triangle . by the cosine law , we have that similarly , consider the , and we have since , we have .now consider the surface delaunay ball that passing through the vertices of the simplex and containing no other points from .recall that is an arbitrary vertex of .since all other vertices of are at least -euclidean distance away from , we then have that the intersection of with the cone contains no point from other than .by applying claim [ claim : twocones ] with , we then obtain that :   in what follows , we will show that a point satisfying the conditions in corollary [ cor : nn ] as well as that indeed exists when .this will then prove the proposition .specifically , we will first identify a sample point , and then we will show that satisfies the requirements of the proposition .  * identifying a point . *let and be two balls tangent to at ; assume without loss of generality that is inside of and is outside .locally around , the surface is sandwiched between and .now consider the plane ; note that also lies in .denote , and .let be the intersection point of and that is on the same side of the line passing through as the point .similarly , let be the intersection point of with on the same side of the line as . obviously , lines on the arc that avoids and .see the right figure for an illustration where the shaded region is .set ; easy to see that .hence we have that now consider the segment and the point such that the length of is a value which we will set later . as is contained in the cone with apex and aperture in the plane , the ball will intersect both segment and , thus intersecting both and . since is inside the surface and outside , it follows that .pick any point . by the -sampling condition of , there must exist a sample point such that .in other words , there is a sample point such that .now set .obviously , the ball ( and thus the point ) is contained inside the cone .observe that by corollary [ cor : nn ] , condition ( c-2 ) is satisfied if ( c-2.a ) ( implying that ) , and ( c-2.b ) ( implying that is contained in the inner cone ) .what remains is to find a value for so that ( c-1 ) , ( c-2.a ) , and ( c-2.b ) are all satisfied simultaneously . note that since , we have that . hence condition ( c-1 ) is satisfied if . since , ( c-1 ) holds as long as the following inequality holds .  since ,if ( c-1 ) holds , then we have that , which implies ( c-2.a ). now consider condition ( c-2.b ) , which holds if .since and , we have that : that is , ( recall that ) . hence condition ( c-2.b ) holds if   putting eqns ( [ eqn : r1 ] ) and ( [ eqn : r2 ] ) together , we have that as long as the value satisfying the following inequality : then conditions ( c-1 ) and ( c-2 ) will be satisfied , and there exists a point as stated in the proposition . given that , we can show that valid exists .for example , for , inequality in eqn ( [ eqn : combined ] ) holds as and

Abstract: the efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points . from a computational viewpoint , the most favored complexes for this purpose have so far been vietoris - rips and witness complexes . while the vietoris - rips complex is simple to compute and is a good vehicle for extracting topology of sampled spaces , its size is huge particularly in high dimensions . the witness complex on the other hand enjoys a smaller size because of a subsampling , but fails to capture the topology in high dimensions unless imposed with extra structures . we investigate a complex called the _ graph induced complex _ that , to some extent , enjoys the advantages of both . it works on a subsample but still retains the power of capturing the topology as the vietoris - rips complex . it only needs a graph connecting the original sample points from which it builds a complex on the subsample thus taming the size considerably . we show that , using the graph induced complex one can ( i ) infer the one dimensional homology of a manifold from a very lean subsample , ( ii ) reconstruct a surface in three dimension from a sparse subsample without computing delaunay triangulations , ( iii ) infer the persistent homology groups of compact sets from a sufficiently dense sample . we provide experimental evidences in support of our theory .


posttranslational modification of proteins is one of the principal mechanisms by which signals are transmitted in living cells . in particular ,covalent modification by phosphorylation is wide - spread and provides one of the basic modes by which a signal is mediated through a pathway .classical signaling pathways typically contain a cascade of post - translational modification cycles ( also called futile cycles ) where the activated protein in one layer acts as the modifier enzyme in the next layer . in order to understand the biological importance of this sophisticated signaling mechanism and to predict the behavior of the system ,a number of theoretical studies have focused on determining the system s dynamics and steady states . in the most discussed case , the mapk cascade , the presence of bistability andoscillatory behavior has been revealed for different choices of system parameters .  the basic features of general signaling cascades can be elucidated from the study of signaling cascades with layers and one cycle of post - translational modification at each layer .these cascades are part of many pathways ( examples include the camp pathway , quorum sensing pathways and the intrinsic cascade in the blood coagulation pathway ) and have been extensively studied from different mathematical points of view : , e.g.  , , e.g.  , and arbitrary , e.g.  . here , we use the classical model of michaelis and menten in which the enzyme - substrate complex is formed reversibly and the dissociation of the complex into a modified substrate and a free enzyme is assumed to be irreversible .further , the modifier , which in the case of phosphorylation typically would be atp , is assumed to be in constant concentration and hence not included explicitly in the model . in thiswe follow much previous work , e.g.  .  in order to simplify the mathematics of the michaelis - menten model ,additional requirements are often imposed ; e.g. the formation of intermediate complexes is ignored under the quasi steady state assumption or the total amount of substrate is assumed to be much higher than those of the ( de)modification enzymes . in the first case , this has the consequence that some biological mechanisms , like sequestration , might be overlooked , while the second case is not , in general , considered biologically valid .additionally , conclusions are often drawn from parameter sampling and are not based on a rigorous mathematical analysis of the model .we focus on understanding the steady state behavior of a signaling cascade with layers and one cycle of post - translational modification at each layer . using mass - action kinetics and without imposing further assumptions on the michaelis - menten model ,we show that the steady states arise as the solutions to one polynomial in one variable .this allows us to conclude that these systems have exactly one steady state for fixed total amounts of substrates and enzymes .this is achieved using an induction argument that reflects the modularity of the cascades .our approach makes it possible to study aspects of the system in detail without relying on simulation .for example , the stimulus - response curve arises as the inverse function of an explicit rational function ( i.e. a quotient of polynomials in one variable ) .several other rational functions are found that describe the species concentrations at steady state ; these functions provide a clear picture of how the steady state re - accommodates to variations in initial concentrations and provide insight into sequestration .in addition , our approach enables us to explore the parameter space in more detail than previously and to divide the space into regions that exhibit different steady state behaviors .the simplicity exposed by the algebraic approach is the key component in this work and follows the line of recent works , as well as some older approaches .all rational functions can be explicitly computed .in addition , their inverses can be obtained using any program that allows symbolic algebraic computations like mathematica .the article is divided into two main sections . in the first the mathematical theory is developed .we show that the steady states can be described as the solutions to a polynomial in one variable only and that there is only one steady state for any initial total amounts of substrates and enzymes .subsequently , we derive rational functions that relate the concentrations of chemical species at steady state . in the second section ,we use these results to discuss various topics of biological interest such as enzyme competition , stimulus - response curves , and how steady state concentrations vary in response to changes in the total amount of substrates .we end with some concluding remarks .proofs are given in the appendix .by a _ rational function _ over a field we understand a function that is given as a quotient of polynomials with coefficients in . assuming that it is in reduced form , that is , there are no common zeros between and , then the only discontinuities of the function are given by the zeros of the denominator .note that two reduced forms and of the same rational function satisfy and for some .if not otherwise stated , we consider rational functions over .if is a finite set ,  ] ( see e.g. ) .there is an evaluation map from  ] , let be the amount of that corresponds to a steady state value of .for example , 90% of the maximal response corresponds to a steady state value of .according to theorem [ stimulus ] , the total amount of enzyme required is determined as .  therefore , our results provide a way to analyze and compute the sensitive or switch - like character of a signaling cascade . in particular , the _ response coefficient _ ( also called cooperativity index ) is given as  , the _ switch value _ as  and the _ hill coefficient _ as  .other measures of sensitivity , probably more suited for an analytical study , escape from our control because the analytical form of the stimulus - response function is required . in our case ,the stimulus is given in terms of the rational function evaluated in the response ( theorem [ stimulus ] ) , hence the stimulus - response curve is the inverse of .we do not have an expression for the inverse and can only evaluate this through tabulation of points .one alternative measure is the _ control curve _ given as , where is the stimulus - response function of the variable  .for , a plot of the control curve could be created similarly to that of by tabulation and using that the derivative is one over the derivative of .further , we have studied the shift in stimulus - response curves for modified substrates in different layers , as suggested in ( * ? ? ?6 ) . for that, we note that is the maximal value of the modified substrate in the -th layer ( proposition [ rationalfunctions ] and theorem [ stimulus ] ) .then , if  ] , where and . since and , then , for all values of for which is defined , and hence it is an increasing function .  in the following we rely on and lemma [ lemma_pos ] . since , we have at steady state . therefore in order for , we need for values of for which holds , we have and , hence , and if and only if .therefore , if holds .define the remaining quantities ( ) through .these quantities are all positive .therefore , a positive value of provides a steady state in if and only if and are satisfied .the question is whether for every value of there is a unique value of satisfying equations and .a.  if , then is satisfied for all values of and so . in this case , there is no positive root of and hence is a continuous increasing function in .b.  if , then is satisfied if .since the only positive root of the denominator of is , we have is a continuous increasing function in .  in both cases , the image of is the set of positive real numbers . by lemma [ rational ] ,the inverse function , , is also a continuous increasing function of in with image set .it follows that there is a unique steady state of the system in for positive total amounts , and .it is straightforward to check that the derivative of as a function of is always positive and hence is increasing , whenever it is defined .recall that , with .the signs of the coefficients of indicate that has either two positive roots or two non - real conjugate complex roots . for , , while for , .therefore , there is at least one positive root .when tends to infinity , becomes positive , and thus the other real root satisfies .it follows that is positive in and and negative in .note that has constant positive sign in and takes negative values in .hence , is positive only if .  if , then meaning that .therefore , any steady state in must satisfy .it is left to the reader to check that the derived steady state values for are in for .we are going to prove the theorem by induction on . for , ( ii )follows directly from proposition [ enzymebehavior ] ; ( i ) follows by composition of in proposition [ enzymebehavior ] with the expression for in .next , assume that the theorem holds for for some .consider the splitting of the cascade of length into the last layer and the first layers . by induction hypothesis , for every value of ,there is exactly one steady state of .in addition , is given by a continuous increasing function of the total amount of substrate in the last layer .hence with a decreasing function of in the interval .further , can be extended continuously to with , since .note that is independent of .on the other hand , we noticed in the main text that is given by a continuous increasing function of , for in the valid interval .the function is zero when and tends to infinity when the root is approached .  therefore , there are two continuous functions of , and , describing the steady state value of .both of them take positive values , but is decreasing , while is increasing in .in addition , and .since the image set of on is , there exists a unique value of for which , and therefore there is a unique bmss .note that the intersection point satisfies , .  what remainsto prove is that the statements ( i ) and ( ii ) also are satisfied for .( i ) the steady state is the unique value for which . note that is a linear polynomial in with coefficients in $ ] .therefore , with .the steady state solution is then given by this function is continuous in the interval , where .in addition , its derivative is positive .indeed , we have that and .also , ( suppressing the subindices ) and .consequently , and is increasing .therefore , by lemma [ rational ] , the intersection point is given by a continuous increasing function of .it proves ( i ) .( ii ) from , which is an increasing function of ; hence by composition of with , is an increasing function of .the inverse of this function is , which is a rational function of , as desired .it is defined on , which is if , and otherwise .the proof of theorem [ onesteadystate ] is completed .the proof is by decreasing induction starting from .the discussion above proposition [ rationalfunctions ] shows that there is a rational function satisfying ( i ) with . to prove that the image is , note that the derivative of with respect to is always positive , and therefore is a continuous increasing function of ( for fixed ) , which tends to infinity as tends to , and vanishes if .( note that only depends on and , so when ( at steady state ) tends to , it implicitely implies that tends to infinity . )( ii ) follows from the expression of , and ( iii ) is likewise fulfilled .now assume the proposition is true for all satisfying for some .consider . by induction hypothesis, are increasing rational functions for all , and so , by , is an increasing rational function of .hence by , is a rational function of given by . if , then is the identity function .note that depends on and , but not on any other total amounts .similarly , depends on and .by induction only depends on and , .  by induction hypothesis , , and are continuous increasing functions of defined on the interval . in addition , as tends to , tends to infinity , tends to the value ( because ) and so tends to .the function is a continuous decreasing function of , ranging from to , defined as the positive root of the polynomial .let .since is increasing as function of , then is decreasing , because is . in order for to be well defined , it must be that ( it is negative otherwise ) .therefore , the possible steady state values of must satisfy .the function is a continuous increasing function that takes the value at ( ) . on the other hand ,when tends to , the function tends to .therefore , there exists a unique value of , , for which .this value is the upper limit for .thus , a necessary condition for valid steady states is that . note that is the -value corresponding to the first positive root of .it follows from that is an increasing rational function of , because the denominator is non - zero for values .it proves the first part of the statement . since for positive values of smaller than denominator is non - zero , we conclude that corresponds to the first positive zero of the denominator .in addition , when tends to , the denominator of tends to zero , while the numerator tends to some positive real number ( since ) and hence tends to infinity showing ( i ) .  to see ( ii ) , note that does not depend on , while does .if is a fixed value of , and , then we have .therefore , the intersection point of the two curves increases if does .in addition , tends to infinity as does ( consider the expression for the positive root ) , and therefore intersection points as close to as desired can be obtained .  from and proposition [ rationalfunctions ] , we have that and are continuous increasing functions of .hence , by the inverse function theorem , is a continuous increasing function of .call this function .it does not depend on .  on the other hand ,the proof of theorem [ onesteadystate ] demonstrates that for every fixed total amount , is a continuous decreasing function of , .it follows from the fact that the total amount in the -th layer , , decreases with .therefore , the steady state value of is the intersection of these two curves , .if now is increased , the first curve does not change , while the second one does .for every fixed value of , we have that if .therefore , the steady state value of increases as does , because we intersect with a growing function .the quantities , , are continuous increasing rational functions of .further , for , the rational functions do not involve .hence , will also be growing with .it follows from that and also increase , while decreases .we proved that an increase of causes to increase and to decrease . using an induction argument ,assume that the statement is true for layers with indices smaller than some value .then , if , by induction hypothesis decreases if increases . therefore , by , decreases and hence increases .it follows that increases , as desired . if , the argument is analogous .since is a continuous increasing function of , we have that , where , is given by a continuous increasing function .additionally , is a continuous increasing function of given by on .we obtain .let be the value of corresponding to , the maximal value of . because , we have . on the other hand , .  because is an increasing function , the proposition follows if .let be the values of corresponding to and , respectively ( zero if ) .since , then by , and the inequality is proved .

Abstract: posttranslational modification of proteins is key in transmission of signals in cells . many signaling pathways contain several layers of modification cycles that mediate and change the signal through the pathway . here , we study a simple signaling cascade consisting of layers of modification cycles , such that the modified protein of one layer acts as modifier in the next layer . assuming mass - action kinetics and taking the formation of intermediate complexes into account , we show that the steady states are solutions to a polynomial in one variable , and in fact that there is exactly one steady state for any given total amounts of substrates and enzymes .   we demonstrate that many steady state concentrations are related through rational functions , which can be found recursively . for example , stimulus - response curves arise as inverse functions to explicit rational functions . we show that the stimulus - response curves of the modified substrates are shifted to the left as we move down the cascade . further , our approach allows us to study enzyme competition , sequestration and how the steady state changes in response to changes in the total amount of substrates .   our approach is essentially algebraic and follows recent trends in the study of posttranslational modification systems .   * keywords * : mass action kinetics , sequestration , futile cycle , post - translational modification , stimulus - response , rational function