rolling adhesion of white blood cells ( leukocytes ) along the vascular endothelium plays a key role in the immune response and is a prominent example for the interplay between transport and specific adhesion in biological systems .random encounters between cell and vessel wall lead to the formation of initial bonds based on adhesion receptors from the selectin - family . because selectins are characterized by fast association and dissociation kinetics , new bonds are readily formed on the downstream side and old bonds are continuously ruptured at the upstream side .if the processes of bond formation and rupture are sufficiently balanced , rolling adhesion results . as rolling velocity is smaller than the velocity of a cell moving freely in hydrodynamic flow , this mechanism allows the leukocytes to more efficiently survey the vessel wall for appropriate molecular signals .the main signals in this context are chemokine molecules , which indicate the presence of an infection and lead to firm adhesion based on the adhesion receptors from the integrin - family .firm adhesion in turn is a prerequisite for transendothelial migration into the surrounding tissue .similar mechanisms are used by stem and cancer cells to disseminate in the body through the blood flow .the main tool for investigating rolling adhesion under controlled conditions are flow chambers .there receptor - carrying cells suspended in an aqueous solution are perfused in linear shear flow above a ligand - coated wall .for example , it was shown with flow chamber experiments that selectin bonds have fast kinetic rates and that the dissociation rate increases with force ( slip bond ) , thus explaining their superior function in mediating rolling adhesion .recently it was demonstrated by a combination of atomic force microscopy and flow chamber experiments that the lifetime of single selectin bonds shows a biphasic response under force .such a catch - slip bond behavior might have evolved to avoid adhesion of leukocytes under static conditions . indeed it has been found early in flow chamber experiments that leukocytes adhere only above a critical threshold of shear .in addition to specific properties of the molecular bonds , transport processes might also play an important role in creating the shear threshold .both rolling adhesion and the shear threshold have also been demonstrated in cell - free flow chamber experiments with ligand - coated micron - sized beads .although cell - free rolling is more erratic than leukocyte rolling , this indicates that cellular features ( like cell deformability ) are not essential to rolling adhesion .flow chamber experiments also allow to study the exact effect of non - molecular parameters , for example of changes in the viscosity of the medium . by adding an inert substance like the sugar ficoll, it is possible to change shear stress but not shear rate in a flow chamber experiment , thus dissecting the respective roles of force and transport .the standard observable in flow chamber experiments is the translational velocity as a function of time . from this time seriesfurther variables can be derived , for example average and standard deviation of velocity . in the case of low ligand density ,a stopped cell is most likely held by a single bond and detaches without rebinding .then the time series can be used to measure single bond lifetime . at higher ligand density, rolling occurs .because rolling is never smooth but erratic due to the stochastic processes on the molecular level , then the time series can be used to measure pause time distributions .the velocity time series can also be used to define different states of motion of the cell .typically a cell is said to undergo rolling whenever its mean velocity ( averaged over some seconds ) significantly decreases compared to the free ( hydrodynamic ) velocity .if no motion can be detected for several seconds , the cell is considered to be in firm adhesion .  because rolling adhesion is not only of large physiological importance , but also characterized by an intricate interplay of different physical factors , it has long been subject to intense modeling efforts .if molecular effects are of interest , then such modeling efforts typically start with physical models for bond association and dissociation .combined with the hydrodynamics of a sphere in front of a wall , they lead to algorithms known as adhesive dynamics .this approach has been applied before to different aspects of rolling adhesion , e.g. , the interplay of two receptor systems or the effect of catch bonds .recently we have developed a new variant of this algorithm which in contrast to earlier approaches fully resolves the spatial positions of the receptors on the sphere and the ligands on the wall . using this approach, we were able to predict the efficiency of initiating cell adhesion in shear flow as a function of the density and geometry of the receptor and ligand patches .a large modeling effort has also been spent on the role of cell deformability and the interaction between multiple particles .here we focus on the case of moderate shear flow and small numbers of cells , when cell deformability and hydrodynamic interactions between cells are not relevant .  a convenient way to present the results from adhesive dynamics simulations of rolling adhesion is the calculation of state diagrams which predict different types of motion over a large range of model parameters . because in flow chamber experiments one usually measures only translational velocity as a function of time , these state diagrams have been determined before based only on the translational velocity simulated as a function of different molecular parameters .however , computer simulations also allow to track the angular velocity of a cell , thus doubling the number of degrees of freedom that could be compared . for example , in physical terms cell rolling means that translation and rotation are synchronized .although this fact was already noted in the pioneering paper on adhesive dynamics , it has not been systematically exploited due to the lack of experimental data . herewe present a detailed analysis of the different dynamic states of rolling adhesion which is based on the simultaneous tracking of both translational and rotational degrees of freedom .in contrast to earlier work in this field , we fully resolve the spatial positions of receptors and ligands and calculate the state diagrams as a function of the on- and off - rate .moreover , we explore the role of different external parameters , including the viscosity of the medium .the paper is organized as follows . inwe explain our numerical method , which is a combination of a langevin equation accounting for hydrodynamic and thermal forces in the limit of small reynolds numbers ( stokesian dynamics ) and adhesive dynamics accounting for the forces caused by the formation of adhesive bonds between cell and substrate . by taking care of the diffusive motion of the sphere ,we are able to explicitly resolve both receptor and ligand positions . as a first application of the hydrodynamic part of our model, we explain the physical difference between slipping and rolling .we close this section with a detailed description of the parameters relevant for the algorithm . inwe point out , using a simple analytical description , how the cell slows down and simultaneously synchronizes its translational and angular velocity under the action of bonds .then we discuss the mean velocities of the cell as a function of the on- and off - rate of the receptor - ligand bonds .based on this we define five distinct states of stationary motion for a cell in a flow chamber .in we identify these different states in state diagrams displaying the dependence on the internal bond parameters , i.e. , the on- and off - rate .in addition , the effect of external parameters to the location of the states in the space of different on- and off - rates is discussed . in the closing section ,, we argue that molecular parameters can be much better extracted from flow chamber experiments if both translational and rotational degrees of freedom are measured , as it is done in our simulations .furthermore we propose experiments based on recent nanotechnological developments that could result in flow chamber data for the angular velocities of cells or microspheres .as a simple model system for a cell or a microsphere in a flow chamber we use a rigid sphere of radius that moves above a wall ( cf . ) . for objects as small as white blood cells ( with diameters m )the typical value for the reynolds number is much less than one and inertia can be neglected ( overdamped motion ) . therefore , the flow around the cell is laminar and well described by the linear stokes equation .cells are usually observed only close to the wall , that is at a distance to the wall which is much smaller than the separation between the two parallel glass plates used in a flow chamber . in this regionthe unperturbed flow profile is approximately linear . throughout this paperwe therefore consider the sphere to flow in simple shear flow , i.e. , we consider the shear rate to be a constant.  in the situation under consideration the sphere is not only driven by the imposed shear flow but also by direct forces exerted on the cell .these include a constant gravitational drift force which results from a slight density difference between the cells and the surrounding medium . for flow chamber experiments, this implies that most cells adhere to the bottom plate , which usually is also the plate coated with ligands . in our context , the strongest forces result after initial binding from receptor - ligand bonds that pull on the sphere . a third source of forceare thermal forces which are ubiquitous for objects in the micrometer range and lead to diffusive motion .this kind of motion can be described by an appropriate langevin equation .it is convenient to introduce a six - dimensional state vector which includes both the three translational and the three rotational degrees of freedom .the first three components of denote the cartesian coordinates with respect to a flow chamber based coordinate system ( cf . ) .the last three components describe the rotation that maps a coordinate system fixed to the center and the orientation of the sphere to the flow chamber fixed coordinate system .similar compact notations are introduced for velocities and forces .the symbol denotes a six - dimensional velocity with the first three components being translational velocities and the last three being angular velocities . is the combined force and torque vector . using this notation , the langevin equation describing the motion of the cell reads  here , is the mobility matrix . is the velocity of the unperturbed shear flow at the position of the center of the sphere . is the shear force , which results from the hydrodynamic interaction between the cell and the wall . denotes all direct forces ( torques ) acting on the sphere , like gravity and bond forces .the terms and describe the effect of thermal noise . is gaussian white noise with here is the ambient temperature and boltzmann s constant .the mobility matrix for a sphere above a wall depends the distance between sphere and wall .thus , the noise is multiplicative , leading to the gradient term in .we interpret the noise in the usual stratonovich sense .however , is written in its it version , as indicated by the superscript .this allows to directly derive a simple euler algorithm for the update step of the configuration of the sphere during a time step . in non - dimensionalform , using the radius of the sphere as the length scale , the inverse shear rate as the time scale , and as the force scale , the first order discretized version of reads here the dimensionless pclet number describes the relative importance of convective versus diffusive motion of the sphere .the first two moments for the gaussian white noise now read the algorithm is also known as stokesian dynamics and has been derived , e.g. , by brady and bossis and for vanishing shear flow by ermak and mccammon . a more detailed description of our algorithm is given in refs . . in order to obtain accurate results when simulating the motion of the sphere it is essential to properly calculate the mobility matrix and the shear force . in our simulations we use a numerical method described by jones et al . which allows to accurately calculate the components of the mobility matrix for a single sphere above a wall for arbitrary sphere wall distances .the values obtained by this method agree very well with the classical results given for some tabulated height values by goldman et al ..  from one readily sees that diffusive motion is less relevant if the pclet number is large . using typical values for white blood cells ( cf . )one finds that is of the order .therefore , regarding the motion in flow direction thermal motion can be neglected on the scale of the cell radius . however , for the motion in -direction the only force on a freely flowing cell is gravitation , which is small due the small density difference between the cell and the surrounding medium and therefore comparable to thermal forces in this direction .for this reason , in our work we use the full langevin equation from .another reason why we keep the diffusion terms in despite the large values has to do with the size of the receptors on the surface of the cell and will become more apparent in the next subsection , .  for the following it is important to consider how the translational velocity ( in -direction ) and the angular velocity ( for rotations about the -axis ) behave in the limits of small and large separation from the wall .if the translational and angular velocities were completely synchronized , the ratio should be equal to unity .this would correspond to rolling in a macroscopic sense , e.g.for a sticky sphere rolling down an inclined plane or for a car wheel on the street .we now show that the situation is different for a cell in free hydrodynamic motion . for denoting the height of the center of the sphere above the wall, denotes the gap between sphere and wall . in we plot and as a function of in the limit of deterministic motion , .  .translational and angular velocity and , respectively , and their ratio are shown as a function of the separation between sphere and wall .the thin lines depict the asymptotic behavior for and , respectively . and are plotted in units of and , respectively . is measured in units of . ]  for the sphere far away from the wall , , we have the case of free flow , thus and .therefore the ratio vanishes in this limit . for the sphere approaching the wall , , both and slowly approach zero due to the hydrodynamic no - slip boundary condition .the ratio however approaches a finite limit as computed by goldman et al .together these results show that hydrodynamic interactions increasingly synchronize translational and angular velocities as the cell approaches the wall . however , the ratio never reaches the value as it would for rolling in a macroscopic sense .therefore , the hydrodynamic coupling between the cell and the wall is not strong enough to lead to rolling and a freely moving cell is always slipping .we now include receptors , ligands and receptor - ligand bonds into our model . throughout this paperwe call the cellular part of a bond _ receptor _ and its counter part on the wall _ ligand_. before a cell receptor can react with a wall ligand to form a complex ( bond ) , there must be a physical transport process which brings the two components to close proximity .formally bond formation can be separated into a transport and a reaction step using the notion of an encounter complex .the encounter complex is formed whenever a receptor and a ligand are less than the capture radius away from each other .therefore , we model receptors as _ reactive patches _ on the sphere surface having a spherical capture range of radius .the capture length bridges the gap between the continuum approach followed by implementing linear hydrodynamics and the discrete nature of the receptor molecules .ligands are modeled as dots on the boundary wall . because we explicitly resolve receptor and ligand positions , it is essential to also include diffusive motion in the algorithm , .although diffusion plays a minor role for the relative position of the cell , it affects the positions of its surface receptors , which are of much smaller size than the sphere itself .  during the time an encountercomplex exists it can react to the final receptor - ligand bond complex with the on - rate .reversely , any bond complex can rupture into the encounter state with the off - rate .experimentally it has been found that the dissociation rate depends on the physical force acting on the bond complex .these early experiments agreed nicely with the simplest model for bond dissociation under force , which had been conceived first by bell : with the dissociation rate at zero pulling force , the force on the bond and the detachment force scale . the bell model , , can be rationalized using kramers theory as being a thermally activated escape over a transition state barrier in the presence of an external force .although recent evidence suggests that the dissociation rate shows a more complicated force dependence at small forces , has been demonstrated to properly describe the dissociation process for selectin bonds in the high force regime .here we use the bell model for conceptual and computational simplicity .  in order to include the probabilistic nature of bond formation and rupture, the algorithm has to be extended to include rules that take care of these processes .such rules have been first set - up by hammer and co - workers and been refined many times to model various aspects of leukocyte rolling .they are now known as _adhesive dynamics_. here , we briefly explain the main idea behind these rules . a detailed description of our implementation of these rules is then given in the appendix , . in order to simulate the motion of a cell under the action of bonds ,the configuration of the sphere is updated at each time step according to .if , after some update , a receptor patch on the surface of the sphere overlaps with a wall ligand a receptor - ligand bond forms with rate , i.e. , the probability for bond formation during time step is .existing bonds are modeled as harmonic springs , i.e. , the force along these bonds is proportional to the bond extension , essentially given by , with and being the receptor and ligand position , respectively ( see ) .force and torque resulting from extended bonds enter then the configuration update equation , , via the direct force term . similarly to bond formation , the probability for bond rupture during time step is given by , where is the force dependent dissociation rate according to .in contrast to earlier implementations of adhesive dynamics , we explicitly resolve both receptors and ligands in space , rather than considering a wall homogeneously coated with ligands at constant density .one immediate advantage of our method is that it avoids a flow rate - dependent rate of bond formation . in order to be able to spatiallyresolve both ligands and receptors , it is necessary to include the brownian motion of the cell .if one considered only the deterministic part of , at low densities of receptors and ligands it could happen that a receptor never finds a ligand . although these extensions of adhesive dynamics lead to increased computational effort , they are closer to real biological systems , which on the molecular level are in permanent thermal motion . in particular , in the future our implementation will allow us to model receptor - ligand kinetics in more molecular detail .as will be discussed later , our algorithm also opens up the perspective to compare adhesive dynamics simulations to flow chamber experiments using substrates with nanopatterned ligand .our model contains thirteen different dimensional parameters . with , , being the natural scales of length , time and force , respectively , we are left with ten numerical ( dimensionless ) parameters appearing in the algorithm .typical values for these ( both the dimensional and dimensionless ) parameters are listed in .the parameters influence the flow properties and besides viscosity we keep these parameters fixed . for the ambient temperature choose room temperature  k. for flow chamber experiments with cells , the choice  k would be more appropriate , but the exact choice of this value , which affects bond dissociation kinetics and effective strength of diffusion , turns out to be irrelevant for the physiologically relevant parameter values chosen here .for the stokes radius of the cells we use m which is about the measured radius of neutrophils , a main type of leukocytes undergoing rolling adhesion ..parameters used for the adhesive dynamics simulations . if no extra symbol for the dimensionless quantity is defined we use the same symbol for both the dimensional and dimensionless representation of this quantity . [ tab : leuk : parameters ] [ cols="<,^,^,^ " , ]   before discussing the impact of the number of receptor patches on the cell surface , we first estimate the number of receptor patches within the contact zone given .supposing the cell touches the wall it immediately follows from basic geometrical considerations that on average receptor patches are in principle in capture range to wall ligands .on the other hand , the average number of ligands in range to receptors is calculated form the product of the projected area of the contact zone to the wall and the ligand density .the radius of the projected contact area is as .thus the number of ligands in the contact zone is .  ina we have and , i.e. , in the contact zone and ( for ) .therefore , the number of receptors limits the maximum number of bonds in all three cases .in addition , using as an estimate for the receptor patch density on the sphere , we have , i.e. , even for receptor patches , they do not cover the contact zone completely .so , not every receptor necessarily encounters a ligand , and therefore the number of tether bonds is even less than .in fact , in the rolling state , we actually measure an average number of for , for , and for existing bonds , respectively , depending on the actual on- and off - rate , which is less than the respective .a shows that the larger , the smaller on - rates are sufficient to support rolling .not every receptor - ligand encounter ( i.e. , overlap of the receptor patch with a ligand ) turns into a bond .this happens only with a probability depending on and the dwelling time of the encounter .thus , the more encounter occur per time the more bonds will be formed at a given rate . as the encounter rate increases with increasing number of receptor patches , this also increases the average number of bonds .  a shows that the transient ii region expands with decreasing .the main effect here is that for smaller the rolling turns into transient motion , while the border line between free and transient motion is much less effected by the decrease in the number of receptor patches .the large -range for the transient states is a signature of few bonds being at work as we will see again when discussing the influence of ligand - ligand distance .then , single tethers slow the cell down ( depending on the off - rate they may either arrest the cell some while , resulting in state ` transient ii ' , or just decelerate them resulting in state ` transient i ' ) , but after dissociation it is unlikely that the current state of motion is supported by further bonds .however , as long as at least two bonds are possible this effect is partly compensated at large on - rates when the probability for receptor - ligand encounter to result in a bond is high .   to demonstrate the impact of ligand - ligand distance ,b shows the state diagrams for and . using the expressions derived in the previous subsection we now have , respectively , for the number of ligands in the contact zone .the mean number of receptors that may form a bonds is .so , in principle limits the number of bonds in the two cases of higher ligand density and is the limiting value only for the very low ligand density .when we measure the average number of bonds at high on - rates and relatively small off - rates ( i.e. in the rolling and firm adhesion region ) we find for the intermediate ligand density with about five bonds , and for the large ligand density with about fourteen bonds on average . the later value is slightly larger than .this can be explained with the elasticity of the bonds that once formed allows them to also exist beyond the contact zone .thus only at very high ligand densities with , the number of receptors limits the maximum number of bonds . for ligand densities with decrease in still leads to an increase in the average number of bonds as this increases the rate of receptor - ligand encounters .the basic effect of decreasing the ligand density as illustrated by b is the shift of all state from the upper left towards the lower right in the on - off state diagram .for example rolling is only supported at larger on - rates and smaller off - rates when the ligand density is decreased ( i.e. , is increased ) . also the border line between the state of free and transient motionis notedly shifted between the two extreme cases of ligand density simulated for b. in fact , the shift of this border line is much more pronounced than in the previous discussed case where the number of receptor patches was reduced .the simple reason for that is that in a is changed by a factor of five whereas in b the is changed by a factor of almost twenty .similarly to the case of reduced number of receptor patches , a small ligand density results in an increased range for transient adhesion .thus , we note again that at a reduced rate of receptor - ligand encounters , rolling tends to be converted to transient motion. we also note that if several parameters are changed at the same time , the overall effect can be qualitatively understood by superimposing the effects of the single parameter changes described above ( data not shown ) .the on - off state diagrams introduced here display the state of motion of a cell or a receptor covered micro - bead for a given set of experimental parameters in a wide range of possible on- and off - rates of the receptor - ligand pair . because the molecular properties of a given receptor - ligand pair can be assumed to be given _ a priori _ , each pair corresponds to one point in the on - off state diagram . in the followingwe show how the considerations regarding the rolling state presented here can be used to determine the dynamic on - rate of a receptor - ligand pair by changing the external parameter . for this line of reasoning the off - rate assumed to be known , e.g. from dynamic force spectroscopy experiments . shows the ratio of the mean velocities of the cell as a function of for different ligand - to - ligand distances .we see that for a given the cell is slipping if the on - rate is small . increasing the on - rate turns the cell s motion into rolling , i.e. , at some critical on - rate the ratio reaches the value which we defined for rolling .the smaller ( larger ) the ligand density ( ligand - to - ligand distance ) the larger is the critical on - rate .more quantitatively this is shown in the inset of where is plotted as a function of .if one now changes the ligand density in a series of flow chamber experiments from low to high values and by doing so determines the -value at which the motion of the cell turns from transient to rolling motion , then one can read of from the inset of the on - rate of the receptor - ligand bond .the described procedure does not work for very small on - rates for which rolling is not possible even for very large ligand densities ( i.e. , ) . on the other hand onecan always find a ligand distance at which rolling is not possible whatever value the on - rate takes .at very low ligand densities the reaction is limited by the encounter rate and even for very large rates rolling is not possible .in this paper we introduced a new version of the adhesive dynamics algorithm .in contrast to earlier work , our approach also includes the diffusive motion of cells resulting from thermal fluctuations .this allowed us to spatially resolve receptors and ligands .an immediate advantage of this approach is that the single bond on - rate can be chosen to be independent of the relative motion of cell and substrate . in this work , we focused on the different dynamic states of motion which can be identified on the cellular level .we first noticed that the action of a single bond not only slows the cell down , but also changes the motion from slipping ( which is the case for cells moving free in hydrodynamic flow ) to rolling in the sense . in the case of multiple bonds , rolling can also be observed regarding the mean values of the velocities , i.e. , at proper rates of bond formation and rupturing .this motivated us to define the state of rolling adhesion as . by extending these observations we defined five distinct states of stationary cell motion .these states were then displayed in so - called on - off state diagrams which showed the impact of different molecular rates .in addition , we investigated the effect of external parameters .for example , we showed that the cellular motion is changed considerably when the viscosity of the medium is changed .our work shows that different dynamic states of cell motion can be defined in a systematic and quantitative way .in particular , calculations of state diagrams allow us to obtain a complete understanding of the way molecular and other parameters determine motion on the scale of a cell .similar approaches have been taken before , but without a proper definition of rolling in the mechanical sense .our computer work now shows that including the rotational degrees of freedom allows us to investigate the transition from hydrodynamic slipping to bond - mediated rolling in a more detailed way .experimentally it is certainly a challenge to obtain similar data in flow chamber experiments for micron - sized beads or cells . in principle , making use of recent nanotechnological developments one could attach receptors to micron - sized beads which are covered with anisotropic surface layers . if these layers are anisotropically reflective , rotational motion of the spheres can be recorded . for cells , one would have to track surface or intracellular markers ( e.g. mitochondria or nuclei ) .we also suggested a new procedure to experimentally determine values for on - rates from monitoring cell motion ( cf . ) .in contrast to the off - rate , which can be measured e.g. from dynamic force spectroscopy experiments , it is very difficult to infer values for the on - rate in cell adhesion , where both partners have to be attached to appropriate surfaces . here, also one could make use of recent developments in nanotechnology . as we have shown above, the ligand - to - ligand distance is a crucial control parameter in our system .recently , is has become possible to control this parameter using nanopatterend and biofunctionalized arrays of gold dots .therefore , in the future our predictions regarding the effect of ligand positioning might be compared directly to experimental data , especially if compared with a measurement of the rotational degrees of freedom .  for conceptual and computational simplicity ,here we have used the bell model for the force dependence of the off - rate . in principle, it would be easy to include more complicated rupture scenarios , like the catch - slip behavior recently reported for both p- and l - selectin .there is good reason to believe that this molecular behavior is essential for the physiological function of these molecules and different theoretical models have been suggested to explain this behavior .if combined with our modeling framework for adhesive dynamics , these models might be tested against experimental data from flow chamber experiments .such an approach would have the big advantage that it avoids testing single molecules outside their physiological environment , which is especially problematic for adhesion receptors which usually are embedded in the plasma membrane and regulated by the cytoskeleton .further possible extensions of our simulation framework include models for cell deformability and hydrodynamic interactions between cells .cell deformations in free flow should become relevant only at shear rates well above 100 hz .in adhesion , cell deformation depends also on the number and strength of adhesion bonds . in the case of strong adhesion, it is well known that also viscoelastic changes occur , including elongation of microvilli . for computational and conceptual simplicity , here we have focused on the case of moderate shear flow and adhesion , when the effect of cell deformability is small .moreover deformations are irrelevant for rigid microbeads , which have been shown to results in similar physical effects as described for cellular systems . in order to combine elastic effects and hydrodynamics ,a very powerful framework is provided by multi - particle collision dynamics , which in principle also would allow to include non - laminar flow conditions and hydrodynamic interactions between cells .however , these effects can be safely avoided in flow chamber experiments by using sufficiently small shear rates and cell numbers .as explained above , an elegant way to test our theoretical predictions experimentally would be the combination of appropriately coated microbeads with nano - structured substrates .finally we comment on the applicability of our approach to other systems that are based on the stochastic interplay between transport and adhesion . in the work presented here ,we have considered a situation where receptor and ligands are both tethered to macroscopic surfaces .however , similar physical processes are relevant if the two molecular binding partners are free in solution , for example in affinity measurements or shear - induced adhesion of blood - clotting factors .another biological system for which the concepts for bond formation , bond rupture and transport discussed here can be applied is the cargo transport by multiple molecular motors .compared with rolling adhesion , the cell is replaced by the cargo ( e.g. , a vesicle ) , the cell - anchored receptors by molecular motors and the substrate - anchored ligands by binding sites on the filament .in contrast to the case of rolling adhesion , now the system is not driven by some external force , but the molecular motors actively step forward and pull their cargo against an external viscous friction force .our approach can also be applied to non - biological systems .for example , it is well known that erratic motion also occurs in the context of sliding friction .a flat slider that is pulled above a plain wall exhibits stick - slip motion in some range of the pulling velocity .it was shown recently that this special kind of motion can be explained assuming the stochastic formation and rupture of molecular bonds between the slider and the wall .thus the stick - slip motion of sliding friction and the erratic movement of cells in rolling adhesions seem to be based on the similar physical principles .this work was supported by the center for modeling and simulation in the biosciences ( bioms ) and the cluster of excellence cell networks at heidelberg .additional support came from the minerva foundation through a short term research grant for cbk .we thank ronen alon and the members of his group for many interesting discussions on rolling adhesion .the sphere s motion is described by .if no bond between the sphere receptors and wall ligands exists , we take for the direct force only gravity into account , i.e. , the six - dimensional force and torque vector is given by with the earth acceleration constant .receptor - ligand bonds lead to additional contributions to both the force- and momentum - part of .more precisely , a bond between a ligand located at and a receptor located at on the sphere s surface ( see ) pulls with a force  is the force extension curve that describes by what force the bond must be pulled to stretch it up to a total length . here , we consider the bonds to be semi - harmonic springs ( cable model ) with the resting length and the spring constant .the cable model is the simplest model for polymeric tethers . in the cable modela bond behaves as a spring only if it is stretched ( extension larger than the resting length ) , otherwise the bond exerts no force on the sphere . treating the receptor - ligand complex as a harmonic spring works fine in the small extension regime . for large extensions the force extension curve for polymersis supposed to grow much faster than linear , and when the bond extension approaches the total contour length of the receptor - ligand complex it even diverges ( strain stiffening ) .however , typical bonds are weak and their rupture probability increases exponentially with force .therefore , we expect bond extensions to be restricted to the linear regime . as the bond force pulls on the sphere s surfacealso a torque results where is the connection vector from the center of the sphere to the point on its surface where the receptor is attached ( see ) .thus , the total force and torque contribution to by the bonds is with the total number of receptors and if the receptor forms a bond and zero otherwise .the are stochastic variables .thus , the contribution lets the direct force also become a stochastic variable .* the sphere s position and orientation is updated according to ( for an explicit description see ref . ) . *the receptor positions in the flow chamber coordinate system are calculated . *each inactive receptor is represented by a capture ball with radius . *if the distance between a receptor and any ligand is a bond is established with probability , then the resting length of the bond is set to the receptor - ligand distance at the instance of bond - formation ( i.e. , the bond force at the moment of bond formation is zero ) and is stored together with the ligand position .a bond can only be formed if the corresponding receptor and ligand are not already part of another bond .* for each active bond , the contribution to is calculated .* each existing bond dissociates with a rate given by the bell equation , .thus , each bond ruptures with probability , where is the instantaneous force acting along this bond .when a bond has ruptured , both the receptor and the ligand can form a new bond in the next time step according to the rule ( iv ) .as for the resting length of a bond always is true , modeling bonds as harmonic springs in both the extension and compression regime would not make much difference to the results that are obtained by the cable model .given the probability for bond formation or rupture or , respectively , a standard monte - carlo technique is used to decide whether the action happens or not : using a pseudo - random number generator a random number from the uniform distribution in the interval $ ] is drawn . if then the respective action takes place , otherwise not .

Abstract: motivated by rolling adhesion of white blood cells in the vasculature , we study how cells move in linear shear flow above a wall to which they can adhere via specific receptor - ligand bonds . our computer simulations are based on a langevin equation accounting for hydrodynamic interactions , thermal fluctuations and adhesive interactions . in contrast to earlier approaches , our model not only includes stochastic rules for the formation and rupture of bonds , but also fully resolves both receptor and ligand positions . we identify five different dynamic states of motion in regard to the translational and angular velocities of the cell . the transitions between the different states are mapped out in a dynamic state diagram as a function of the rates for bond formation and rupture . for example , as the cell starts to adhere under the action of bonds , its translational and angular velocities become synchronized and the dynamic state changes from slipping to rolling . we also investigate the effect of non - molecular parameters . in particular , we find that an increase in viscosity of the medium leads to a characteristic expansion of the region of stable rolling to the expense of the region of firm adhesion , but not to the expense of the regions of free or transient motion . our results can be used in an inverse approach to determine single bond parameters from flow chamber data on rolling adhesion .


graph isomorphism problem , or exact graph matching , which has numerous applications , is the problem of determining whether two given graphs are isomorphic .that is , there is a bijection between vertices of two graphs which preserves adjacency and non - adjacency of vertices . in order to solve this problem, one could consider each of the bijections between vertices of two graphs , and check whether it is an isomorphism between the two graphs .if they are not isomorphic , one would need to check all bijections to realize this fact .but , even for relatively small values of , the number is unmanageably large .hence , finding an effective method with tolerable computational complexity to distinguish the isomorphism or non - isomorphism of two graphs is very valuable .whilst polynomial - time isomorphism - testing algorithms have been found for special classes of graphs , such as trees , planner graphs , graphs of bounded degree and ordered graphs , but no polynomial runtime algorithm is known for this problem .that is , in the worst case the computational complexity of any available algorithm is exponential in the number of graph vertices .one approach to solve this problem is using graph spectrum , i.e. the eigenvalues of matrix representation of a graph , as a graph invariant . since, the matrix representation of a simple graph is symmetric , graph spectrum is computable in time .thus , a graph spectrum can be regarded as a fast graph invariant to distinguish graph isomorphism classes , provided that it has been chosen properly . from this point of view ,a good matrix representation for graphs is a matrix that produces minimum possible number of cospectral graphs .this idea widely has been used and investigated in several papers , such as spectral determination , spectral characterization , spectrum as a graph invariant , comparing graphs via graph spectrum , graph clustering .  until now, several alternative matrix representations for graphs have been proposed in the literatures , such as adjacency matrix , laplacian matrix , signless laplacian , normalized laplacian and heat kernel .the spectrum of all these representations may be used to characterize graphs .wilson and zue have compared these spectrums according to cospectrality . they have shown that there are some graphs which even the combination of these spectrums is not able to separate them .in other words , even the combination of these spectra is not sufficient to determine the graph isomorphism classes , uniquely . to avoid cospectral graphs , emms et al. introduced a matrix representation inspired by the notion of coined quantum walk .they have shown the graph spectrum with respect to this matrix is successful to separate many known families of cospectral graphs .but , there are some non - isomorphic graphs , like a 14-vertex graphs pair , which are not distinguishable by their spectrum . in addition, the size of the matrix that they have proposed is in the order of .  here, we proposes a new matrix representation for graphs .the eigenvalues of this matrix , as a graph invariant , provides a simple polynomial - time algorithm to efficiently solve the graph isomorphism problem in practice .continuous spectrum and split technique which are defined in this paper , mutually , provide a fast computable graph invariant to solve some computationally hard problems , such as graph isomorphism and graph automorphism , in practice .the continuity matrix representation for graphs is defined in the next section .the eigenvalues of this matrix which is in terms of a variable is called continuous spectrum .this spectrum is equipped with split method in the third section to make it more efficient in separating non - isomorphic graphs .the paper is concluded in the fourth section .some derived relations and theorems about continuous spectrum are given in the appendix .in this section , we define continuity matrix representation of graphs . also , we describe why we expect the eigenvalues of such matrix to behave well in separating non - isomorphic graphs .+ at first , let us review some used definitions and notations .the adjacency matrix of a graph is a matrix $ ] where if is adjacent to and otherwise .the laplacian matrix of a graph is .matrix is diagonal degree matrix , whose diagonal elements are given by the vertex degrees .the sign - less laplacian and the normalized laplacian matrix , respectively , are and . + for a given spectrum , two graphs are cospectral if they share the same spectrum .a graph is said to be determined by its spectrum , or ds , if any graph that is cospectral with is also isomorphic to .let be the adjacency matrix of an -vertex graph and be a fixed non - negative integer .we define continuity matrix where and is a real variable . is a diagonal matrix with the diagonal entries of matrix .continuous spectrum of is the eigenvalues of .  clearly , the matrix is the same adjacency matrix , if or .continuous spectrum is the eigenvalues of matrix and , consequently , is in terms of . to compute the continuous spectrum for a given graph , it is sufficient to assign an arbitrary value to .computer results show that small value of is enough to separate non - isomorphic graphs by the continuous spectrum .for instance , taking is enough to discriminate graph isomorphism classes for graphs with up to vertices . as an example , the continuity matrix representation of the graph fig.[graf ] is given ( for ) , here .  a sample graph and its continuity matrix , width=151 ]  \ ] ]  continuous spectrum digram in terms of , width=453 ]  figure [ diagram ] depicts the continuous spectrum of the mentioned graph in terms of .since this graph is bipartite , its diagram is symmetric about the origin .the proof of this property for bipartite graphs is given in the appendix .the aim of defining such spectrum is to enhance the ability of graph spectrum in distinguishing non - isomorphic graphs .we would like a graph spectrum to play the role of a fast computable invariant to discriminate non- isomorphic graphs .let us see why we expect continuous spectrum to be efficient in discriminating non - isomorphic graphs .let be the adjacency matrix of a graph and is an eigenvalue of with the corresponding eigenvector .thus , we have . in other words , the above system of equations says that the summation of the values of neighbors of a vertex should be the value of that vertex multiplied by . in this view, is a possible value under this condition that summation of neighbors values of each vertex should be the value of that vertex multiplied by .thus , only the neighbor vertices are regarded and other vertices are ignored in each equation .+ a natural question arises here .why only the direct neighbors are considered and other vertices which are two or more steps away are ignored ? how can we consider the effect of vertices which are not neighbors of vertex in the -th equation of the above system of equations while preserving the importance of the neighbors ?+ therefore , we want to modify adjacency matrix to take into account the effect of non - neighbor vertices while preserving the importance of the neighbors .we use an attenuator factor , namely , to reduce the effect of far vertices .hence , in each equation we enter the effect of vertices which are steps away , by adding their corresponding entries multiplied by . also , if there are more than one way between and , we enumerate all of them .exactly speaking , we consider -length walks and their numbers .we know that -entry of indicates the number of -length walks between and .thus , we take into account the effect of vertex which is edges away from by -entry of multiplied by . then , we put zero on the diagonal .therefore , the desired matrix is what you see in the above definition .graphs which are determined by their spectrum , abbreviated by ds , are those that do not have any cospectral mate . in order to compare the successfulness of one spectrum in characterizing a set of graphs , we count the number of graphs which are determined by that spectrum .the distinguishing ability of adjacency spectrum , laplacian , signless laplacian and normalized laplacian spectrum are compared with continuous spectrum in table [ small ] .[ smallgraphs ]  .[ small ] [ cols="^,^,^,^,^,^,^",options="header " , ]   we have measured the distinguishing ability of split spectrum with respect to different graph spectra by the fraction of ds graphs that they produce in table [ srg ] . we know that the eigenvalues of all known graph representing matrices , such as adjacency matrix , laplacian matrix and even continuity matrix is the same for strongly regular graphs with the same parameters .but , as computer results show split continuous spectrum is able to easily distinguishes the isomorphism classes of such graphs . + in table[ srg ] , the results of using split spectrum for discriminating graph isomorphism classes of srgs are shown .this table shows that the split spectrum with respect to adjacency matrix is not able to successfully separate non - isomorphic classes of srg graphs .in contrast , the combination of continuous spectrum with split technique provides an efficient tool to distinguish graph isomorphism classes . as we see in the table , taking and , is enough to discriminate all mentioned instances of strongly regular graphs with to vertices .  since the computational complexity of this method is for -vertex graphs , it significantly outperforms to other methods which are exponential .thus , it is a valuable approach to solve graph isomorphism problem in practice , even for strongly regular graphs .the eigenvalue of a matrix representation of graphs , i.e. graph spectrum , is invariant to arrangement of graph vertices and is computable in .therefore , it is a good invariant to separate non - isomorphic graphs .but , usual spectrums such as adjacency spectrum , laplacian spectrum and even combination of them suffer from existing cospectral graphs .for example , for all strongly regular graphs with the same parameter , both adjacency and laplacian matrix produce the same eigenvalues .  in this paper , a new matrix representation for graphs , i.e. continuity matrix , is proposed .this representation has improved the ability of graph spectrum in recognizing non - isomorphic graphs .the definition of continuity matrix is based on the extension of the equations that eigenvalues satisfy in . in a graph adjacency matrix , each equation is established for a vertex and its neighbors , while in the new defined matrix each equation is written for all vertices .this fact makes the graph spectrum more engaged in graph structure and , consequently , enhances its ability in discrimination of non - isomorphic graphs .therefore , finding cospectral graphs with respect to this new matrix is difficult .  for discrimination of some special graphs such as strongly regular graphs , which have been known as hard cases of the isomorphism problem , continuous spectrum is not sufficient , alone .thus , split spectrum was proposed to improve continuous spectrum in separating non - isomorphic graphs .we have shown by computations that split continuous spectrum , which is computable in , successfully characterizes the isomorphism classes of many known families of strongly regular graphs up to 64 vertices .continuous spectrum and split method , mutually , provide an efficient invariant to discriminate non - isomorphic graphs .such graph spectrum which is highly entangled with graph structure benefits us in solving some computationally hard problems , such as graph isomorphism and graph automorphism problem , in practice .split continuous spectrum , which does not need any heuristic attempt , provides a simple , efficient and fast graph invariant to distinguish graph isomorphism classes in average time and in worst case .i would like to thank ted spence and richard wilson for providing graph data .10  lszl babai . on the complexity of canonical labeling of strongly regular graphs ., 9(1):212216 , 1980 .lszl babai , xi chen , xiaorui sun , shang - hua teng , and john wilmes .faster canonical forms for strongly regular graphs . in _foundations of computer science ( focs ) , 2013 ieee 54th annual symposium on _ , pages 157166 .ieee , 2013 .lszl babai and paolo codenotti .isomorhism of hypergraphs of low rank in moderately exponential time . in _ foundations of computer science , 2008 .ieee 49th annual ieee symposium on _ , pages 667676 .ieee , 2008 .xiao bai , richard c. wilson , and edwin r. hancock .manifold embedding of graphs using the heat kernel . in _ proceedings of the 11th i m a international conference on mathematics of surfaces _ , ima05 , pages 3449 , berlin , heidelberg , 2005 .springer - verlag .n. biggs . .cambridge university press , 2nd edition , 1993 .strongly regular graphs , partial geometries and partially balanced designs . , 13(2):389419 , 1963 .fan r. k. chung ., volume 92 of _ cbms regional conference series in mathematics_. published for the conference board of the mathematical sciences , washington , dc , 1997 .drago cvetkovi and slobodan simi .graph spectra in computer science . , 434(6):15451562 , 2011 .drago cvetkovi , slobodan k simi , and zoran stani .spectral determination of graphs whose components are paths and cycles . , 59(12):38493857 , 2010 .david emms , edwin r. hancock , simone severini , and richard c. wilson . a matrix representation of graphs and its spectrum as a graph invariant ., 13(1):research paper 34 , 14 pp .( electronic ) , 2006 .chris godsil and gordon royle ., volume 207 of _ graduate texts in mathematics_. springer - verlag , new york , 2001 .willem h. haemers and edward spence .enumeration of cospectral graphs ., 25(2):199211 , 2004 .john e hopcroft and jin - kue wong .linear time algorithm for isomorphism of planar graphs ( preliminary report ) . in _ proceedings of the sixth annual acm symposium on theory of computing _ , pages 172184 .acm , 1974 .xiaoyi jiang and horst bunke .optimal quadratic - time isomorphism of ordered graphs . , 32(7):1273 1283 , 1999 .david s johnson .the np - completeness column ., 1(1):160176 , 2005 .paul j kelly et al . a congruence theorem for trees . , 7(1):961968 , 1957 .fenjin liu and qiongxiang huang . on the spectral characterization of -shape trees ., 61(3):355367 , 2013 .eugene m. luks .isomorphism of graphs of bounded valence can be tested in polynomial time ., 25(1):42 65 , 1982 .b. mckay .https://cs.anu.edu.au/people/brendan.mckay/data/graphs.html .ronald c read and derek g corneil . the graph isomorphism disease ., 1(4):339363 , 1977 .ted spence .es / srgraphs.html .daniel a spielman .faster isomorphism testing of strongly regular graphs . in _ proceedings of the twenty - eighth annual acm symposium on theory of computing _, pages 576584 .acm , 1996 .edwin r. van dam and willem h. haemers . which graphs are determined by their spectrum ? , 373:241272 , 2003special issue on the combinatorial matrix theory conference ( pohang , 2002 ) .richard c wilson and ping zhu .a study of graph spectra for comparing graphs and trees ., 41(9):28332841 , 2008 .the graph shown in fig . [ graf ] is a bipartite graph and the digram of its continuous spectrum , as we see in fig .[ diagram ] is symmetric about the origin . in other words , in bipartite graphsthe eigenvalues of and for any arbitrary is the same , but with a negative sign .the next theorem proves this fact .    * proof * : it is sufficient to show that if is an eigenvalue of then is an eigenvalue of and vice versa .assume the continuity matrix of is where and are square matrices corresponding to each part of bipartite graph . clearly , these two matrices are odd functions of . and the functions and are even functions of . + let the partitioned vector be an eigenvector of with eigenvalue where and relate to parts and .now , is an eigenvector of with eigenvalue , because + .+ it follows that and are respectively eigenvalues of and with the same multiplicity .    * proof * : it is sufficient to take in the previous theorem .+ there are several theorems for ordinary graph spectrum which are remained also true for continuous spectrum .some of them are given here .their proof is similar to what is given in algebraic graph theory books for adjacency spectrum , such as .let be the continuity matrix of graph , and a permutation of .then is an automorphism of if and only if , where is the permutation matrix representing .let be a simple eigenvalue of for a specified , and let be a corresponding eigenvector(with real components ) . if the permutation matrix represents an automorphism of then .

Abstract: the graph isomorphism problem is a main problem which has numerous applications in different fields . thus , finding an efficient and easy to implement method to discriminate non - isomorphic graphs is valuable . in this paper , a new method is introduced which is very simple and easy to implement , but very efficient in discriminating non - isomorphic graphs , in practice . this method does not need any heuristic attempt and based on the eigenvalues of a new matrix representation for graphs . it , almost always , separates non - isomorphic -vertex graphs in time and in worst cases such as strongly regular graphs , in time . here , we show that this method , successfully , characterizes the isomorphism classes of studied instances of strongly regular graphs ( up to 64 vertices ) . strongly regular graphs are believed to be hard cases of the graph isomorphism problem .


during the last few years there has been a lot of activity in the study of networks once it was realised that networks of diverse nature exhibit many common features in their underlying structure .the most important property that appeared to be commonly occurring in such networks is the small world property .this means that if any two nodes in the network is separated by an average number of steps , then , where is the total number of nodes in the network . in some networks , even slower variation ( i.e. , sub - logarithmic )scaling has been observed .the first indication that networks have small world behaviour emerged from an experimental study by milgram , in which it was shown that any two persons ( in the usa ) can be connected by an average number of six steps . following the tremendous interest in the study of networks, a new experiment has been done to verify this property in real social networks . some studies which involve simulations on real networks have been made also .parallely , the question of navigation on small world networks has been addressed theoretically in many model networks .it must be noted that it is not necessary that a navigation or searching on a small world network would show the small world property , i.e. , the dynamic paths may not scale as .this is because searching is done using local information only while the average shortest distances are calculated using the global knowledge of the network .this was explicitly shown by kleinberg in a theoretical study where nodes where placed on a two dimensional euclidean space .each node here has connections to its nearest neighbours as well as to neighbours at a distance with probability .although the network is globally a small world for a range of values of , navigation on such networks using greedy algorithm showed a small world behaviour only at .in general the path length showed a sublinear power law increase with .recently , the scale - free property of networks has also been found in many real - world networks which precisely means that the degree distribution follows a behaviour , with the value of between 2 and 3 for most networks .navigation on such networks has also been theoretically studied yielding the result that rather than logarithmic , there is a power - law variation of the path lengths with as in the non - scale free euclidean network ( except for particular algorithms or values of ) .this confirms once again that for navigation in a small world network , it is not necessary that the path lengths will reflect the small world behaviour .obviously , a lot depends on the algorithm also , but any realistic algorithm is basically local . for euclidean networks , the greedy algorithm appears to be the most popular one from the findings of the original milgram experiment as well as that of . here ,once the source and the target nodes are chosen , the strategy is to connect to a neighbour closest to the target . in scale - free barabsi - albert ( ba ) networks , the two extreme strategies of a random search and highest degree search have been used as well as a preferential search scheme based on the degree . strategies involving other properties like betweenness centrality etc .have also been investigated .a relevant question in this context is how to test the quality of the search strategy . noting that the path lengths in all the studies show a behaviour , one may expect that a lower value of indicates a better search strategy .  in realistic searches as in milgram s experiments and that described in , the search may not be successful or complete always .( in fact for graphs which are not fully connected , there is always a finite possibility that a connected path does not exist , even with global knowledge . ) here , apart from the search length , the success rate is also an important factor . indeed , most works referring to milgram s experiments ignore the fact that very few chains were completed in the initial experiment ( 3 out of 60 ) and better success rate ( 35% or more ) was achieved only after randomness in the search was compromised in the sense that the choice of the source persons was made with some bias .many of the theoretical models do not consider the probability of termination at all and the interest is to find out whatever the search length is .we argue that for realistic searches , where termination of search paths is a possibility , one must consider both the success rate as well as the path lengths to adjudge the quality of a search strategy .we have generated random scale free networks and used a tunable preferential search strategy which can be extended from a random search ( rs ) to a highest degree search ( hds ) scheme .in general we find that , the success rate also follows a power law behaviour such that . on a scale - free network, it is expected that a high degree search will improve the search quality .however , we have shown that a comparison of search strategies based on the behaviour of path lengths alone may lead to an entirely different conclusion .on the other hand , we find that the ratio of the success rate to the path length , has a power law behaviour with , i.e. , where decreases as one conducts a higher degree search ( section ii ) .we therefore claim that is a reliable measure to test the quality of a realistic search on a network involving both the success rate and the shortest path lengths in an effective manner .calculation of on simple networks suggests that can take up values between zero and one ( section iii ) .corresponding to we have the best searchability ; this we call a `` dynamical small world '' ( dsw ) effect . in section iv, we have shown that is indeed sensitive to the distinguishing features of networks using the examples of different kinds of scale - free networks with the same degree exponent .finally , in section v we have considered a social network in which the nodes possess a `` similarity '' factor and the the linking probability of two nodes depends parametrically on it . here , we have shown that it is possible to obtain dsw ( i.e. , assumes values very close to zero ) for a finite range of values of the parameter .in this section we describe the searching procedure on random scale - free ( rsf ) networks , which have no other features other than being scale - free .random scale - free networks are generated by assigning the degree to each node following a scale - free distribution , allowing to vary from to . is taken to be while is allowed to vary from 1 to higher values .the links are then established randomly between the nodes with the given distribution .links are assigned in steps ; we start with the node with the highest degree , in the next step the node with the second highest degree and so on .this method has been previously used to generate random scale - free networks .the search strategy is like this : + a source node and a target node are selected randomly .the source node will send the signal to one of its neighbouring nodes provided that node has not already taken part in the search .this is in tune with milgram - like experiments .-1 cm  if one of its neighbour happens to be the target itself , the message will be send to the target . if not , then the neighbour will receive the message with a probability , where and is the degree of the node .this continues and if it happens that the message can not be passed any more , the chain will remain incomplete .we take the average dynamic shortest path as the average of all completed chain lengths .obviously corresponds to a random search while gives rise to the highest degree search . were considered for the ba network in , although the algorithm was slightly different .we first report the results for and for for a network of nodes with different values of .the path length against shows a very interesting result .normally one would expect that the path length will reduce with higher degree searches . while this happens for 3 or 4 , for smaller values of , this is not true . here ,on the contrary , the path length increases with ( fig .for all values of , we notice that the path lengths saturate to a constant value which corresponds to the highest degree search .this occurs for finite values of .the fraction of successful attempts shows a uniform behaviour with ; for all values of , it increases with ( fig .1b ) as expected .the reason why for small there is a difference in behaviour of is quite obvious .when is small and a random search is carried on , very few chains can be completed , but if completed , these are essentially small in length as the connectivity on an average is small . on making a higher degree search , more chains can be completed but at the expense of increased path lengths .the static small world effect is quantatively measured by the behaviour of the average shortest path lengths ( global ) with .we notice that the dynamic path lengths varies with and in all cases it shows a power law increase with ( except for and where it has a logarithmoc increase ) .however , for small , the path length against shows that it varies as with increasing with a higher degree search , which is counter intuitive ( fig .so clearly the path length alone is not a good measure of the searchability .the success rate , on the other hand , shows a change in behaviour as is varied .the exponent shows a decrease with ; for small values of , is positive and it goes on to assume negative values for higher .however , when we study the behaviour of the ratio as a function of , we find a consistent behaviour , where decreases as is increased ( fig .3a ) .  for higher values of , path length and success ratesshow expected behaviour with ; both improve with . also decreases with monotonically as in the case of ( fig 3b ) .hence one can regard as a measure to adjudge the quality of the search strategy in general . for higher values of ( which implies higher connectivity ) , variation of with not that appreciable showing that for networks with larger connectivity , the sensitivity to degree - based algorithm is not remarkable .the searchability factor and hence can be calculated in some simple cases .although the examples may seem too idealised , the results help us to understand the bounds of .search on a regular network with nearest neighbour interactions only .the simplest case is a one dimensional chain ( fig .4a ) . on any euclidean network , a greedy allgorithm , i.e., passing the message to a node closest to the target is meaningful . hereas the links are between nearest neighbours only , the search path length is equal to where is the distance separating them . in general where .the success rate is hundred percent here such that and in this case .note that in this case , since the message can not be passed to the same node more than once , the search must take a unique path , it has to be directed towards the target in steps of 1 ( shown by the arrows in fig 4a ) .thus random search is not conceivable here .if the message can be passed to the same node more than once , one can use a random search and the situation is analogous to a random walker .the path length will now be proportional to and the corresponding .this is in fact the worst case scenario : however , it does not belong to the class of navigation considered here where messages can be received only once .search on a tree with uniform branching ratio : let us imagine a tree with branching ratio ( fig .4b ) and with levels .let us consider the case of sending a message from the top ( zeroeth level ) to a node at the bottom ( ) level .the probability that a message is sent to this node at the level from the top is . noting that the number of nodes , implying here .more generalised cases where search from arbitrary sources and targets are considered may give different values of .for example , if one considers search paths of length 1 only , then the success rate is giving .larger paths will have a different and in principle can vary from 0 to 2 .if one takes all the possibilities , i.e. , all possible pairs of source and targets , then the average can be estimated from the following expression : where is the ratio for two nodes separated by edges and the number of such pairs .the terms in the numerator decrease with ; for , , , while and .thus the leading order term in the numerator is while the leading order term in the denominator is giving for very large trees .search on a gel - here we consider a network in the form of a gel or a star ( fig .4c ) where there is a central node to which all other nodes ( having only one link ) are attached .again path lengths are of the order 1 and success is a surity . hence here .. a chain of hubs or gels :  we consider a idealised chain of hubs or gels , it contains hubs with degree each ( fig .4d ) ; the total number of nodes .now a search path length will be of the order of , say . if we consider a random search , the probability of completing the search is . or, the thermodynamic limit may correspond to two different cases .the first is when is small but is finite .the search probability is small ( but non - zero ) and does not vary with ( as the ration remains finite ) giving .  on the other hand if is large and finite , such that goes exponentially to zero rather than as a power law .the above discussion shows that when , the value of for small world networks lies between and when the search is done randomly .obviously , for a more efficient algorithm is lesser than that for the random search and thus it can be said that for searches in which repetition of messengers is not allowed , .a dynamic small world in this perspective is one in which .  in this section ,the simple networks which have been considered have no loops and hence the paths from one node to another is unique .thus the important quantity determining is essentially the success rate . in the trivial cases like the linear chain and the gel, even the success rate is deterministic . on more complicated networks , both and will have to be computed probabilistically .in this section we show that our measure of searchability is sensitive to the distinguishing features of different networks .we have considered networks which are equivalent as far as the degree distribution is concerened in the sense they are all scale - free with the same exponent but differ in certain other aspects .we have selected scale - free networks with the same exponent and used a hds algorithm which is meaningful in a scale - free network . in the two subsectionwe give a brief desciption of the models we have considered and the results obtained .we have considered three different types of scale free models :  \1 .the barabsi albert ( ba ) network :  the ba network is grown using a preferential attachment scheme . here , we have started with a single node and nodes are attached to the network one by one .an incoming node gets attached to the existing node with the probability where is the degree of the node at that time .this network has a has a tree structure as we allow only one link to an incoming node .the random scale free ( rsf ) network :  this network is the same as that described in section ii . we have taken here and kept such that it is comparable to the ba network .scale free assortative network ( sfa )  the third network which we have considered is a scale - free model with tunable assortativity .this scale - free assortative ( sfa ) network is generated by modifying slightly the method of generating the rsf network . to each node , the degree is assigned from a scale - free distribution as in rsf .let and be the degrees assigned to nodes and . while establishing the links of the node , attachment to the node will be definitely made if .otherwise they will be linked with a probability here controls the assortativity factor .for , there is no assortativity and for  , the network is assortative ( disassortative ) .it may be noted that both the rsf and the ba networks have zero assortativity .we employ the hds algorithm , i.e , a node always passes on the information to the neighbour with the highest degree . in the ba network , for system sizes considered here , shows a power law increase with with an exponent which becomes smaller with showing that it approaches a logarithmic increase with for large .this is to be expected as it has a tree structure .( in it was shown that even with loops , the highest degree search scheme in the ba network results in a logarithmic increase of the path lengths with . ) follows a power law decay with such that .  for the rsf network, we again apply hds and find that in this case , although the path length varies logarithmically with .( it may be mentioned here that for the rsf , as is increased from 2 to 3 , the searchability becomes less efficient in terms of and also becomes less sensitive to the factor . )thus in terms of simply the path lengths the rsf and ba networks are equally searchable which is counterintuitive . on the other hand a more realistic result .  in the third network , even when it is weakly assortative ( ) , we find that shows a deviation from the random network value . here which means that it is more searchable than the rsf but less than the ba network with the same algorithm .the behaviour of with for different networks is shown in fig .as already mentioned , in the ba as well as in the rsf networks , the path length shows a logarithmic increase with but since is appreciably greater than zero , neither of these two networks is an ideal dynamic small world .the random scale - free network has a worse searchability because of the sharper decay of the success rate . in the sfa network , there is a tendency of clustering of high degree nodes thereby allowing loops .hence the path length does not scale logarithmically with .however , one only needs to know the behaviour of to comment on the searchability and here it shows a slower decay with compared to the rsf network although the magnitude of is lesser .till now we have discussed search strategies on network which are basically characterised by their degree distribution .social networks , which are found to be scale - free in many cases , have some additional features .it has been found that in a society , people have different characteristics and the links depend on these bonds .these characteristics can be related to profession , hobbies , geographical locations etc .we have considered a very simplified picture with one such characteristic which we call the similarity factor of the individuals varying between 0 and 1 randomly . in a scale - free network with given distribution , we first assign the degrees as well as the similarity factors to the nodes .the bonding between two nodes , subject to this distribution is now made according to   thus we will have a scale - free network of degree exponent in which similar nodes will try to link up for .the similarity factors can be looked upon as coordinates in a one dimensional lattice in which case the network is scale - free network as well as euclidean .  in the earlier casesonly degree based strategies have been considered as the networks did not have any other characteristic feature .but as shown in such a strategy is rather artificial for social networks .here we have considered two different strategies ; one is the standard hds while the other one is a greedy algorithm based on the similarity factor . in the latter, a node sends the signal to a neighbour which is most similar to the destination node .we call this the highest similarity search ( hss ) .we investigate how the searching strategies are dependent on based on the value of .we consider networks with and here and vary to obtain networks varying on the basis of similarity based bondage between the nodes for the same degree exponent .we have shown in section iv that the measure is able to distinguish between networks of same degree exponent which differ in other characteristics and hence it is meaningful to employ this measure to investigate the searchability based on different algorithms here .again we find that shows a power law decay with for both strategies in general .however , for the highest degree search , the value of shows marginal dependence on and the network is moderately searchable with this strategy as lies between 0.5 and 0.6 . using the hss on the other hand we find an appreciable variation of with .in fact , we find that for , the network is highly searchable as is here . for , becomes lesser for higher values indicating that it might go to zero in the thermodynamic limit giving rise to a truly dynamical small world effect .it is true that not all social networks are scale - free .however , our studies lead to the important result that _ even for scale - free networks _ , if there is a social distance dependent linking scheme , the degree based search is rather inefficient ( compared to the one based on the social characteristic ) .thus we conclude that for any social network , the degree based search will be less useful .  herewe have considered networks of size with a cutoff in the degree distribution at ( ) . in order to simulate the friendship or acquintance networks , where in principle ,the whole population of the earth is involved , it is better to use a degree distribution with an exponential cutoff or simply exponential decay to make the study more realistic .in this work we have proposed a new method to study the searchability on networks where the search may not be complete always as transmission of signals can not be repeated to the same node . in real searches ,failure to complete a chain may be due to various other reasons .the motivation comes from milgram - 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Abstract: we consider navigation or search schemes on networks which are realistic in the sense that not all search chains can be completed . we show that the quantity , where is the average dynamic shortest distance and the success rate of completion of a search , is a consistent measure for the quality of a search strategy . taking the example of realistic searches on scale - free networks , we find that scales with the system size as , where decreases as the searching strategy is improved . this measure is also shown to be sensitive to the distintinguishing characteristics of networks . in this new approach , a dynamic small world ( dsw ) effect is said to exist when . we show that such a dsw indeed exists in social networks in which the linking probability is dependent on social distances .