with the advent of the square kilometre array ( ska ) , magnetic fields will be regularly probed out to high redshift .it is necessary , therefore , to model theoretically such magnetic fields .much recent progress has been made in magnetohydrodynamical ( mhd ) cosmological simulations , but the dynamic range allowed by such simulations is still too small to faithfully account for important ( sub)galactic - scale processes including dynamo action .therefore , theoretical modelling of magnetic fields of galaxies , using a combination of analytical and numerical approaches , is crucial . in particular , magnetic fields of magnitude known to be present in the diffuse interstellar media ( ism ) of spiral galaxies .such fields can be loosely divided into large - scale ( also known as regular ) fields , which are coherent on scales larger than those of turbulence , and small - scale ( also known as fluctuating ) fields , which have coherence lengths of at most the outer scale of turbulence , .a typical estimate for within the disc of a galaxy is ( * ? ? ?* hereafter ) .  in the presence of cosmic ray electrons ,the large - scale magnetic fields of galaxies are sources of polarized synchrotron emission .the component of the field parallel to the direction of propagation of such emission causes the plane of polarization to rotate via the faraday effect .likewise , polarized emission from background sources such as active galactic nuclei ( agn ) will undergo faraday rotation as it passes through intervening galaxies on its way to the observer . therefore ,modelling the large - scale magnetic fields of galaxies may be useful for predicting and interpreting cosmological radio polarization and rm observations ( e.g. * ? ? ? * ) .realistic galactic magnetic field models could also be useful for helping to constrain the environments of fast radio bursts ( frbs ) , using their observed rms ( e.g. * ? ? ?* ) , or modelling the transport of cosmic rays of extra - galactic origin from within their source galaxies .at present there is no analytical solution for galactic large - scale magnetic fields that is sufficiently generic , realistic , and versatile to be suitable for the kinds of applications outlined above .phenomenological models are undoubtedly useful for some problems , but models that are more physically motivated , yet still consistent with observations , are needed .one approach to painting magnetic fields onto galaxies is to evolve each galactic field separately using a mean - field dynamo simulation .such simulations would use as input evolving parameters from , e.g. , a semi - analytical galaxy formation model , .however , this approach is computationally rather demanding .an alternative , albeit less rigorous approach , is to assume that such dynamo processes lead to saturation on timescales small compared with galaxy evolution timescales , allowing one to adopt a steady - state solution for the magnetic field using as input coeval galactic parameter values. an analytical solution ( * ? ? ?* hereafter ) has indeed been used for just this purpose , but this solution contains no information about the vertical distribution of magnetic field within the galaxy . on the other hand , the 3-dimensional , or at least ` 2.5-dimensional ' ( 2.5d , cylindrical symmetry ) , spatial structure of the fieldis required for some of the applications mentioned above .this motivates the main purpose of the present work : to show how such a 2.5d solution of the suitably approximated dynamo equations can be constructed , and to demonstrate the efficacy of this solution by comparing it with numerical solutions of a less approximate set of dynamo equations .our model leads to a steady quadrupole - like configuration for the regular magnetic field in the saturated state , which is consistent with results from standard galactic dynamo theory .dynamos in thin accretion discs are probably more complicated , with , e.g. , oscillatory solutions often obtained , and our model is not meant to be applied in such a context . on the other handit is not meant to describe any galaxy in particular .we realize , however , that modelling the field of our own galaxy , in particular ( e.g. * ? ? ?* ; * ? ? ?* ; * ? ? ?* ; * ? ? ?* ) , is important for many investigations , such as modelling cosmic ray propagation or subtracting foreground emission in cosmological studies . in this context, our model can perhaps serve as a step toward more physically - motivated galaxy models .the paper is organized as follows . in section[ sec : model ] , we motivate the full set of dynamo equations , discuss the numerical method used to solve them , and summarize the model for the underlying galaxy .this is followed by an explanation of the analytical method in section [ sec : analytic ] .our main results are presented in section [ sec : results ] .here we compare analytical and numerical solutions for four different parameter regimes .further , for each parameter regime , we compare a suite of numerical solutions , each obtained from a successively more approximate set of equations .thus , a secondary aim is to shed light on the applicability of various approximations used in the literature .the implications of our results are discussed in section [ sec : discussion ] , and we offer some conclusions in section [ sec : conclusions ] .following the standard prescription , the magnetic field and velocity field are each written as the sum of a mean ( denoted by uppercase ) and a fluctuating ( denoted by lowercase ) component .averaging the induction equation we then obtain the standard result where we have neglected terms involving the microscopic ( ohmic ) diffusivity since the magnetic reynolds number in galaxies . here is the mean electromotive force , where ` bar ' denotes mean . for isotropic turbulence, can be approximated by the expression  where can be written as the sum of kinetic and magnetic contributions , with and . here is the correlation time of the fluctuating flow and , with the density .other terms in equation  may not always be negligible ( e.g. * ? ? ?* ) , but as our main aim is to present a basic analytical solution , we leave such complications for future work .the turbulent magnetic diffusivity is estimated as the kinetic term , meanwhile , is estimated as with where is the angular velocity . in our models , so . the expression for is the standard formula of krause , while the expression for includes the effects of ` rotational saturation ' . for , and be rotationally quenched , but that case does not arise in our models .the evolution of is governed by the dynamical quenching equation , where is the equipartition field strength .we have neglected an ohmic term in equation , which is anyway negligible for realistic values of the -flux density .we assume for simplicity that ; that is , we assume a strouhal number . the strength of the mean magnetic field in the saturated state is approximately proportional to and thus to .  in general a flux density of the form is considered , where the advective flux density is given by , and the diffusive flux density by , with the turbulent diffusivity of . exploring the influence of other potentially important contributions to the helicity flux is left for future work .we do , however , compare our results with results using the simple algebraic -quenching formalism , with a parameter which in the literature has typically been set to unity .we make use of a generalized algebraic quenching formalism that allows to be estimated analytically from the dynamical quenching equation .it is first convenient to define the following dimensionless parameters : where is the density scale height ( disc semi - thickness ) and is the vertical mean velocity at . in words, is the radial shear parameter , is the dimensionless scale - height , is the dimensionless vertical mean velocity , and is the ratio of turbulent diffusivities of and .the parameter in the generalized algebraic quenching equation  is then estimated as    as we are dealing with axially symmetric magnetic fields , it is convenient to express the magnetic field in terms of scalar potentials and . here , the flux function enters through the poloidal field , , and for the toroidal potential .then ( alternatively , we could have used the variables and , where is the mean vector potential ( e.g. * ? ? ?* ; * ? ? ?here and below , cylindrical polar coordinates ( , , ) are used , with the galactic angular velocity along the -direction , and at the galactic midplane .  for simplicity , we assume that mean radial velocities vanish , that is independent of , and that turbulent diffusivities are constant , but the general equations are provided in appendix [ sec : full_equations ] .the toroidal and poloidal parts of equation  and equation  can then respectively be written as  \biggr.\\ &    \biggl.-\eta\left[\frac{{\partial}\psi}{{\partial}r}\frac{{\partial}t}{{\partial}r }          -t\lambda^-\psi + \frac{{\partial}\psi}{{\partial}z}\frac{{\partial}t}{{\partial}z}\right]\biggr\ }    -\frac{{\partial}}{{\partial}z}({{{u}}_z}\alpha{_\mathrm{m}})\\ &    + \kappa\lambda^+\alpha{_\mathrm{m } } ,  \end{split}\ ] ] where , and is generally given by equation , but sometimes by equation .horizontal and vertical boundary conditions must be chosen for the variables , and . evidently , at .we further demand that , by symmetry , as ; this then implies at .finiteness of implies ; we choose without loss of generality .this is the natural choice since is then proportional to the magnetic flux through a horizontal disc of radius centred at the position , finiteness of implies .note that , while , as , and these values of and are set explicitly .we also set to avoid a singularity in the diffusive flux term at the origin .further , we adopt the same boundary conditions at , corresponding to the outermost radius of the simulation domain ; solutions are insensitive to the choice of boundary conditions at since is chosen to be well outside the region of dynamo action .we set .we impose vacuum boundary conditions at , which implies at .thus , the mean magnetic field is forced to be vertical outside the disc . in imposing these boundary conditionswe ignore the gaseous halo , leaving its inclusion for future work . to see how these boundary conditions arise , begin with the requirement that outside the disc .this implies .obviously , as , which then implies at .setting also leads to the condition .the right - hand side is usually small compared with the left hand side , so this condition can be approximated by outside the disc .this approximation becomes invalid for , so our model is less reliable at the very centre of the disc .since as we must have at .further , we set at , which allows to flow across the disc boundary .  we have also experimented with other choices of boundary conditions on .we tried various combinations of ( instead of ) , so that ( instead of ) and so that ( instead of ) at .for example , choosing implies .interestingly , for models with a strong outflow , changing the boundary conditions in any of these ways has almost no effect on the solution . however , for models without an outflow ( or with a weak outflow ) , solutions are rather sensitive to the vertical boundary conditions .the explanation for this is that if the outflow removes the field and at the boundaries rapidly enough , boundary effects can not propagate inward .changing the boundary condition on to at has little effect on the solutions for , even if no outflow is present .however , if is adopted at the vertical boundaries , then undergoes a sign change in boundary layers near the disc surface , leading to magnetic field configurations that are rather different from the standard solutions ( for models without a strong outflow ) . on the other hand , if we choose instead of at the boundaries , we find that either the code does not converge or , if at is also chosen , the field decays .we use an grid that is linear in both coordinates , and solve the equations using the same finite differencing ( 6th order ) and runge - kutta time - stepping ( 3rd order ) routines employed in the pencil code . for thin disc solutions , we use a resolution , excluding ghost zones , for the runs presented , while for the model with a thicker disc , we use .a much smaller resolution , e.g. , is usually sufficient for testing purposes . runs with even higher resolutionwere performed as a check for convergence , when necessary .initial conditions at time are {{\rm e}^{-z^2/h^2}}$ ] and .the form of the seed field is chosen so as to be relatively simple and to satisfy the boundary conditions , though solutions do not depend on initial conditions as long as the seed field is sufficiently small .although our dynamo model includes the lorentz force via the dynamical quenching non - linearity , we assume that the dynamo parameters ( with the exception of ) do not evolve with time , e.g. in response to the evolution of the mean magnetic field . as we are mainly concerned with steady - state solutions in the saturated regime , this is not an important limitation .below we describe the model of the underlying galaxy within which the dynamo operates .the mean velocity field takes the form ( , , ) in cylindrical coordinates , with .we employ a brandt rotation curve of the form where and are parameters .note that is a maximum at , and that approaches a constant value as .the vertical velocity is given by with the amplitude a parameter which is equal to the magnitude of the vertical velocity at the disc boundary .we adopt an exponential profile for the equipartition energy , where is the equipartition field strength at , and we set the radial scale length ( e.g. * ? ? ?below , magnetic fields are quoted in the arbitrary unit .table [ tab : models ] lists the various parameter sets ( ` models ' ) explored in this work . for simplicity ,we adopt constant values of , , , and .all models employ a disc with constant turbulent velocity .model a , with , and , is the fiducial model .model b differs from model a in having a nonzero outflow velocity , model c differs from model a in having a twice larger value of the turbulent scale , and thus also of , while model d differs from model a in having a twice larger value of the disc half - thickness ..parameter values for our four models .all models employ a disc of constant turbulent speed .the fiducial model , model a , lacks an outflow and assumes a turbulent outer scale and disc semi - thickness .model b differs from the fiducial model by having a nonzero vertical outflow speed at the disc surface .model c has no outflow , but assumes , while model d is like model a but uses .dimensionless parameters , and defined in equations  are also provided .[ cols= " < , < , < , < ,< , < , < " , ]   ( 13.4,2.3)   we first summarize the local critical no- solution . to begin with , the slab mean induction equation reduces to a set of - and -independent algebraic equations , parameterized by the radius . these equations can be solved to yield vertical averages for and , while can be estimated from the condition . for the kinematic regime at least ,the local solutions can be extended to the global domain using wkbj theory ; we use this method below to estimate the global growth rate of the mean magnetic field .  in terms of the parameters ,the _ local _ exponential growth rate in the kinematic regime is given by , & \mbox { if } { \mathrm{co}}\le1;\\   \sqrt{\frac{2q}{\pi}}\frac{\omega}{h }    \left[\frac{1}{\sqrt{{\mathrm{co } } } }        -\frac{1}{12}\sqrt{\frac{\pi}{2q}}\left(\frac{\pi^2 + 6hv}{h{\mathrm{co}}}\right)\right ] , & \mbox { if } { \mathrm{co}}>1 .\end{dcases}\ ] ] this is the growth rate that obtains , at a given , if radial transport terms are not included .another quantity of interest is the magnetic pitch angle , with . in the saturated state , , with and in the kinematic regime with   to obtain the magnetic energy density in the saturated state , we must make use of equation  for .we then obtain , where and where the normalized dynamo number is given by note that must be greater than unity for a supercritical ( ) dynamo .that is , the dynamo number in the kinematic regime must exceed in magnitude the critical dynamo number , which is given by note that for , we obtain the convenient result .we also note in passing that to write down these no- solutions , certain numerical coefficients had to be calibrated to numerical solutions , but for our purposes these can now be thought of as fixed ; here we use the expressions from .in the kinematic regime ( ) we have , where is the growth rate of the fastest growing global eigenmode .to estimate the global growth rate , the standard procedure is to separate a suitably simplified version of the mean - field induction equation  into two equations : one equation for the local ( -dependent ) part and another for the global ( -dependent , or , in general , - and -dependent ) part .here we consider the case of a spatially constant turbulent diffusivity .the schrdinger - type global equation that results from this analysis can then be solved approximately using wkbj theory ( e.g. * ? ? ?this leads to an eigen condition for the fastest growing mode , ^{1/2}dr=\frac{\pi}{2},\ ] ] where the ` potential ' is given by the ` energy ' eigenvalue by and where the integration limits in equation , and , are the zeros of the integrand . equation  with equations  and is solved by numerical iteration to yield .one could also calculate the kinematic eigenfunction under the wkbj approximation , but as we are mainly interested in saturated solutions , we do not attempt this .below we summarize the other type of analytical solution needed to construct our hybrid analytical solution : the local critical perturbation solution .like the no- solution , the perturbation solution contains a parameteric dependence on , but unlike the no- solution it retains a dependence on .however the local critical perturbation solution is not sufficient on its own because it is unnormalized .  in terms of the parameters , solutions for the field components and in the saturated state are ,  \end{split}\ ] ] and ,  \end{split}\ ] ] where }\right\}^2.\ ] ] and is a parameter which can be estimated using the local critical no- solution , as explained in section [ sec : combined ] .note that in equation  is slightly different from obtained from the no- approximation in equation , which is why the former has been labelled with a superscript .what has not been done previously is to combine these two types of analytical solution for in the saturated state to obtain a normalized axisymmetric solution that depends on both ( parametrically ) and . to do this, one must use the no- solution to estimate the parameter , hence normalizing the perturbation solution .we simply set where is obtained from equation , and here and are the magnetic field components of expressions  and normalized by .the ` ' in equation  comes from the sign invariance of the induction equation , and the sign must be chosen to match that of the numerical solution , which ultimately depends on the arbitrary seed field chosen .note also that is proportional to , which varies exponentially with in our model , so to remove this -dependence , we adopt the midplane value of in equation .this choice is partly motivated by the good _ a posteriori _ agreement we obtain with numerical solutions ( section [ sec : results ] ) .thus , equations and , with equations , , and comprise the analytical solutions for and in the saturated state . to estimate , we make use of the property . then , in cylindrical coordinates we have ',\ ] ] where is the integration variable , and where we have made use of the fact that for quadrupolar solutions .the prescription presented above is rather trivial , as it relies simply on combining , in a straightforward way , two known analytical solutions . yetit is potentially powerful because it generates a magnetic field solution that is effectively global and , though it is 2.5d , can be utilized in a 3d context .below we present examples of these analytical solutions and compare them with numerical solutions under various levels of approximation .the different types of approximation are summarized in table [ tab : approximations ] , and discussed further below.   +  +   +  +   +   +   +   +   +   +   +   +   +   +   +   +   +   +we now present full numerical solutions for the galactic disc dynamo , and compare these with numerical solutions obtained under various approximations , as well as with the analytical solution discussed in section [ sec : analytic ] .we consider four cases , labelled as models a to d , which have the parameters listed in table [ tab : models ] . the fiducial model , model a , has no outflow and has standard parameters , and .model b differs from model a only in that it has a fairly strong outflow , with ( for comparison , the field decays if ) .model c , on the other hand , has the same parameters as the fiducial model except that .this causes for , which means that is rotationally saturated ( equation ) in model c for . for other models , decreases monotonically with .it reaches a maximum amplitude of at for models a and b , and for model d , while for model c , has a constant amplitude of between and .model d is like model a except that is doubled form to .disc dynamo solutions are shown in figures [ fig : bcompare][fig : bicontour_d ] .figure [ fig : bcompare ] shows data that is vertically averaged across the disc , with each column of panels corresponding to a given model , from left to right : a , b , and c , while figure [ fig : bcompare_d ] shows the same information for model d. figures [ fig : bicontour_a][fig : bicontour_d ] show the full spatial dependence of the mean field in the steady state for models a d .axes are shown to scale to emphasize the aspect ratios , and the -axis is truncated for presentational convenience . to begin withwe consider three quantities : the kinematic growth rate of the mean magnetic field , and the energy and pitch angle of the mean magnetic field in the saturated state .the time evolution of the square - root of the mean magnetic energy density over the simulation domain is plotted in panels ( a ) , ( b ) , and ( c ) of figure [ fig : bcompare ] and panel ( a ) of figure [ fig : bcompare_d ] .this quantity is normalized to the equipartition value at the origin .the magnetic energy initially decreases with time for the seed field adopted , but after a few hundred the fastest growing eigenmode begins to dominate and the field then grows exponentially .the final solution is not sensitive to the seed field chosen as long as it is sufficiently weak compared to equipartition .the magnetic energy saturates after the start of the simulation , after growing orders of magnitude , depending on the model .the global growth rate in the kinematic regime is for model a , for model b , for model c , and for model d. this corresponds to ` -folding ' times of , , , and , respectively .the analytical estimate for , obtained using equation  and shown by a dashed line , is for model a , which is remarkably close to the actual value . for model b , however , the analytical estimate is , or about of the actual value .the underestimate for the large case is not surprising , and can be traced to inaccuracies introduced by making the no- approximation on terms containing . in particular, the no- solution can not be simultaneously calibrated to give accurate pitch angles _ and _ accurate growth rates for strong outflows . for model c ,the analytical estimate is , which is also remarkably close to the actual value , while for model d the wkbj treatment yields , or larger than the actual value .analytical solutions rely on approximating the disc as thin , so are expected to be less accurate for model d , which has a thicker disc .   in panels [ fig : bcompare](d)(f ) and[ fig : bcompare_d](b ) , the dashed lines show the normalized magnetic energy density averaged across the disc , plotted against radius , in the saturated regime ( ) .the normalized equipartition field is shown as a dotted line for reference .the field is of approximately equipartition strength , and a few times larger for model a than for model b , as the stronger outflow in model b leads to a less efficient dynamo .this , in turn , is explained by referring back to equations  and . written in terms of , where represents or , and concerning ourselves only with the first terms on the right hand side , we have the first term on the right - hand side is the expansion term , and leads to a reduction of in our model , whereas the second term is the advection term and leads to an enhancement of .though both terms are important , the expansion term turns out to be larger , leading to an overall reduction of .for the same reason , the saturated magnetic energy tapers off more rapidly with radius for model b than for model a. the magnetic field of model c , meanwhile , saturates at a strength a few times larger than that of model a. note that ( orange ) dominates over ( blue ) and ( green ) in all models , as expected , except at , where may exceed and ( see also figures [ fig : bicontour_a][fig : bicontour_d ] ) . for model d ,the vertically averaged magnetic energy actually peaks at , due to the strong vertical field there , but is much weaker for other models .  the analytical solution  for is shown as a dashed - dotted line in panels [ fig : bcompare](d)(f ) and [ fig : bcompare_d](b ) , and generally gives a remarkably accurate estimate of the saturation field strength . in model c ,the agreement is slightly worse for smaller than the radius at which the field strength peaks .this is caused by the neglect of the effect in the analytical model ; when this effect is excluded from the mean - field simulation , the agreement is much better .the effect is more important for larger or , which was also found in .therefore , the increased importance of the effect with seems to persist even when rotational saturation of is implemented .note also that the analytical solution , being local , does not taken into account the radial transport .therefore , it can not be applied at radii or where and are the radii for which the normalized dynamo number , and where the analytical field strength goes to zero in each plot . moreover , the domain for which is generally underestimated by the analytical solution .this is because the solution overestimates the damaging effect of the outflow in the kinematic regime ( as mentioned above when discussing ) , and also because the effect and terms involving are ignored in the analytical solution .we now turn to the two - dimensional plots of figures [ fig : bicontour_a][fig : bicontour_d ] , which show the field components for the saturated state , under various approximations , described in section [ sec : approx ] .we begin by comparing the full numerical solutions of panels ( a ) for models a d .the obvious difference in going from model a to the strong outflow model b , is that the mean magnetic field becomes confined to a smaller radial domain , because of the damaging effects of the outflow .the component is still peaked at about , where the dynamo number peaks , but the components and , whose generation depends on the -effect , are peaked at somewhat larger radii compared to model a because the outflow weakens the dynamo at small . in model c ,the peak of the dynamo number occurs farther out at about due to the rotational saturation of , and the peak of is located at about the same radius .the components and are also pushed out to larger radius compared to models a and b. these features can also be seen in the middle row of figure [ fig : bcompare ] .we have also run a variation of model c without rotational saturation , but do not show the figures for the sake of brevity .as expected , the field in that case resembles qualitatively that of model a at small radius , e.g. in that its components respectively peak at roughly the same radii as in model a. however , the profile of retains the ` double - tail ' morphology at small radius ( dip at the midplane ) seen in figure [ fig : bicontour_c ] , as well as the long tail extending out to . in the thicker disc model , model d ,striking features are apparent at small radius , including a strong double - tail morphology in . at , changes sign at about , which is not very different from model a , in spite of the twice larger in model d. however , real galaxies probably typically have equal to a few hundred at , flaring to for  , which makes the small- behaviour of model d less relevant . at larger radius , the field resembles qualitatively that of model a , though scaled to occupy the larger cross - section of the disc .more interesting , perhaps , is that is less extended with radius than in model a , while is more extended .importantly , the overall field strength falls off less slowly with radius than for model a because remains above unity even at large .   in panels ( g)(i ) of figure[ fig : bcompare ] and ( c ) of figure [ fig : bcompare_d ] , we plot the mean magnetic pitch angle averaged over the disc cross - section in the saturated ( solid ) and kinematic ( dashed ) regimes .the spikes at in figure [ fig : bcompare](i ) and the peculiar behaviour of for are caused by wrapping around from to , as per the definition .when averaging over the disc cross section , these discontinuities also lead to unphysical wiggles in over the finite grid .this is just a consequence of the discontinuous definition of , not of any numerical problem in the simulation . if were redefined to allow values , would continue to rise smoothly as .note also that the unphysical behaviour of at in panel [ fig : bcompare](h ) is not of any consequence as the field is effectively zero in that region .as expected , , is of the order of for the fiducial model , model a , and is slightly larger when an outflow is present , in model b. also as expected , in the saturated regime is generally smaller than that in the kinematic regime .blue dashed - dotted and dotted lines show the predictions of the analytical model for the saturated ( equation ) and kinematic ( equation ) regimes , respectively .note that the analytical prediction for in the kinematic regime is independent of , whereas the numerical solution shows a weak dependence of on in the kinematic regime , as can be seen by comparing the dashed lines in panels ( g ) and ( h ) of figure [ fig : bcompare ] .analytical solutions predict remarkably accurately , especially for the saturated state .this is not completely surprising as they were calibrated against local numerical saturated solutions .however , at small in model c is larger than predicted because of the effect . nevertheless , the agreement with global numerical solutions is striking and implies that the analytical solution and the approximations underlying it are quite robust .of course , solutions will in general depend on the galactic rotation curve , and rotation curves that vary more rapidly in radius than the one adopted here would lead to larger deviations of the global solutions from the local estimates , as the latter ignore radial derivatives .we study the effects of the various approximations using figures [ fig : bicontour_a][fig : bicontour_d ] ; a concise summary of these approximations can be found in table [ tab : approximations ] .we have also studied plots of residuals between , , and from any two different panels of a given figure , but for the sake of brevity , these are not shown .below , we sometimes refer to models run using different approximations as ` sub - models ' ; in going from one panel ( sub - model ) to the next in sequence , we make one additional approximation but otherwise keep the sub - model the same .panels ( a ) show the full solution .we first compare panels ( a ) with panels ( b ) , which show the solution .this approximation neglects terms containing in equation . for modela , shown in figure [ fig : bicontour_a ] , the effect leads to a slight increase in the field components away from the midplane for , which is noticeable in the slight dip in at .the same effect is seen in model b , but is less pronounced , while for model c , the effect is visually similar , but here the effect produces a decrement in near the midplane , with an associated ` push ' of to slightly larger radius . for model d ,the field strength at small is greatly reduced in panel [ fig : bicontour_d](b ) compared to [ fig : bicontour_d](a ) , and the effect on the morphology of is qualitatively the same as for model a , but more pronounced .  in going from panels ( b ) to ( c ) of figures [ fig : bicontour_a][fig : bicontour_d ] , is now neglected in the equations , and then computed _ a posteriori _ using .this is known as the thin - disc approximation , and is formally valid for .terms involving in the mean induction equation are _ not _ neglected in this approximation , however , as from . in our implementation , this means neglecting radial derivatives of , _ except _ in equation .in fact , in going from panels ( b ) to ( c ) equations  and are not affected ( we confirm that kinematic solutions for and are almost identical ) , so any difference between the saturated solutions of these panels stems from the non - linearity . for models a c , there is a slight weakening of the field at small radius in going from panels ( b ) to ( c ) .we note that the magnitude of , in particular , is significantly reduced at , though its spatial distribution is only weakly affected .still , the thin - disc approximation can be seen to be rather robust for these models .however , for the thick disc model , model d , departures are more significant , as expected .  in going from panels ( c ) to ( d ) of figures [ fig : bicontour_a][fig :bicontour_d ] , radial derivatives are now neglected : the local or slab approximation .this leads to rather drastic cutoffs in and at small and large , where the local solutions are subcritical .these cutoffs are of course unphysical .the sharp cutoffs in also lead to a poor reconstruction of .it is also worth noting that peaks at a slightly larger value when the slab approximation is made .nevertheless , these plots show that the slab approximation is quite valid for , for the parameter values considered .perhaps fortuitously , slab solutions of panels ( d ) better reproduce the strength of , near its peak , than do thin - disc solutions of panels ( c ) .the stronger in panels ( d ) as compared to panels ( c ) is mainly due to the neglect of the radial diffusion term in equation  in the slab approximation : re - running submodels ( c ) with this term omitted leads to stronger comparable with that of submodels ( b ) and ( d ) .we next consider the effect of adopting the generalized algebraic quenching non - linearity in place of the dynamical quenching formalism .that is we use equation  with equation , where evaluates to , , , and for models a , b , c , and d , respectively .thus we compare panels ( d ) , which , like panels ( a)(c ) , use dynamical quenching , with panels ( e ) , which show solutions for a sub - model identical to that shown in panels ( d ) , except now using generalized algebraic quenching .the effect on the magnitude and morphology of is seen to be rather small . for model bwe see that the distributions of and broaden with respect to the midplane .further , ceases to changes sign at the disc surface , in agreement with the findings of .finally , we compare the above solutions with the analytical solutions of panels ( f ) of figures [ fig : bicontour_a][fig : bicontour_d ] , calculated using the same resolution as that of the numerical grid .that solution was obtained by normalizing the critical perturbation solution using the critical no- solution , as explained in section [ sec : combined ] and table [ tab : approximations ] .we have already seen from figure [ fig : bcompare ] that analytical no- solutions reproduce rather faithfully the vertically averaged properties of the full solutions .how well do 2.5d ` no- + perturbation ' solutions approximate full numerical solutions ?it can be seen that for all four models , the analytical solution of panels ( f ) of figures [ fig : bicontour_a][fig : bicontour_d ] agrees reasonably well with the full numerical solution of panels ( a ) , but fail at small radius .the differences between analytical and numerical solutions are more noticeable for model b than for models a and c , because of the limitations of the analytical methods for the case where large vertical outflows are present . in the presence of a strong outflow ,analytical solutions underestimate the radial extent of the magnetic field .for the case with vanishing outflow , and presumably for weak outflows , the analytical solution is as accurate as the numerical slab solution , shown in panels ( d ) , which neglects radial derivatives .we note that standard estimates of the outflow velocity give of the order of  , so in this context the outflow in our model b of is very strong . for model d ,the thicker disk causes departures from the numerical solutions to be larger than for model a , as expected .the technique we have used to obtain a 2.5d analytical solution is simple ( even seemingly naive ) .yet it gives rather accurate results , so long as we are not concerned with the very centre of the disc and so long as outflows are not very strong .why should this be ?   more specifically , how could it be that radially local axisymmetric solutions , obtained at each radius by neglecting radial derivatives , can effectively be pasted together to form a global solution ?the field saturates first at the radius approximately corresponding to the peak of the eigenfunction in the linear regime , and then proceeds to spread out .this spreading does not happen mainly by radial diffusion , but by the much faster local growth of the field .therefore , saturation at each radius takes place almost independently .the exception to this picture is what takes place near the boundary of the dynamo - active region , i.e. at the radii and , where .there the local growth rate becomes negative and non - local radial processes become crucial for reversing the balance , and rendering the dynamo supercritical .unfortunately , there does not seem to be a simple way to model the field saturation for and , so the field cuts off steeply ( and unphysically ) at and .any improvement to the analytical model to allow for a more gradual tapering off of the field , in better agreement with numerical solutions of panels ( a)(c ) of figures [ fig : bicontour_a][fig : bicontour_d ] , would be usefull .   more broadly, our results explicitly demonstrate the viability of various approximations often used in the galactic dynamo literature , while exposing some of their limitations .the approximation is shown to be reasonable for typical parameters , but leads to an underestimate of at small radius , for instance .for the rotationally saturated case of model c , and thicker disc of model d , the solution is inaccurate at small radius , where is important .making , in addition , the thin disc approximation , for which terms are neglected in the simulation , and calculated _ post factum _ , does not lead to a large difference in solutions for models a c , which have a thin disc , but is significantly reduced in magnitude . for model d , which has a thicker disc , such departures are even more significant , as would be expected .the slab approximation pastes together local solutions that depend only parametrically on .as explained above , this solution is invalid near the radii where the local growth rate transitions from positive to negative , but is otherwise remarkably accurate . replacing the dynamical quenching non - linearity by a generalized form of the algebraic -quenching non - linearity , which is relatively easy to implement and less demanding computationally ,is seen to have very little effect on the slab solution . finally , the analytical solution , which combines no- and perturbation solutions , is surprisingly accurate , though not without limitations , as discussed in sections [ sec : results ] and [ sec : local ] .as soon as one resorts to pasting together local solutions , the solution for the very central region of the disk becomes invalid .this is far from ideal , but not particularly damaging , for at least three reasons .first , the other approximations , such as the approximation , also lead to inaccuracies in this region .even the vacuum boundary conditions imposed are rather unjustified there , as discussed in section [ sec : boundary ] .further , other effects which have been neglected , such as the presence of the galactic bulge , or a small bar in sab galaxies , for example , would also be more important near the centre . in summarythen , we are probably losing less information than it seems in the analytical model by ` losing ' the very centre .second , for potential applications , such as estimating the faraday rotation due to intervening galaxies on distant agn , the surface area would effectively enter the calculation .as of the radius corresponds to only of the surface area , the effect of the very centre is likely to be negligible , given other potential uncertainties in the problem .third , ionized galactic discs are likely to be flared , with typically somewhat smaller than at the centre .this would allow the validity of all of the sub - models presented to extend down to a smaller radius .for the same reason , the analytical model also underestimates the field at larger radius .the field in this region is relatively weak , though the relative surface area is significant .the neglect of the field in this region is probably only a concern for models with strong outflows , since the disparity is greater in that case , as discussed in section [ sec : results ] .we see then that the analytical solution , though crude , is reasonably accurate .it reproduces faithfully certain key properties of the solution , such as the local vertically averaged pitch angle , as well as the overall magnitude of each field component .what makes this solution potentially powerful , however , compared with more accurate numerical solutions , is the ease with which it can be implemented .such solutions could be painted onto all of the galaxies in a cosmological simulation almost instantaneously and almost effortlessly , to any desired spatial resolution .the greatest shortcoming of such solutions may be their neglect of non - local effects in the disc .on the other hand , the various parameters can be made to depend on without complicating the method .for example , making the disc flared so that depends on is a rather trivial extension .it may also be worth pointing out that analytical solutions are far more transparent , with regard to their interpretation , than their numerical counterparts .the disc solutions presented can be seen as ` basic ' solutions for spiral galaxies , and they _ may _ be rather sufficient to explain the regular magnetic fields of _ certain _ galaxies . however , many galaxies harbour strong , apparently large - scale , magnetic fields high above the ionized thin disc . to include this part of the galaxy in our models ,it is necessary to relax certain approximations , such as the -independence of , , etc ., that may be reasonably accurate for a disc , but fail drastically for a disc - halo system .such an extension to our models is a logical next step , especially considering that haloes , being relatively extended , may be important in the faraday rotation of background sources .important work using mean - field dynamo simulations to explore large - scale magnetic fields of disc - halo systems has been carried out , and provides a useful starting point .at the same time , analytical dynamo solutions for the halo are at present unavailable and would be interesting to develop .ultimately , the aim would be to model analytically the disc - halo system , and compare solutions with more realistic numerical solutions , as done above for the pure disc case .we see the present work as a necessary ` stepping stone ' on the way to this goal .we have presented a simple , effectively 2.5d axisymmetric analytical solution for the large - scale magnetic field of a disc galaxy in the saturated state .this solution was shown to agree with full numerical solutions to a reasonable level of fidelity , with the agreement extending down to the magnitudes and spatial morphologies of each individual component of the field .  where our analytical solution is notably less reliable than full numerical solutions is at small radius , or smaller .it is also less reliable if vertical outflow speeds from the disc are larger than , or in the upper range of , standard estimates , or if the disc is thicker than standard estimates in the inner few .our solution is simple , transparent , and takes virtually no time to implement .it is also versatile because the underlying parameters , such as the turbulent speed of energy carrying eddies , disc scale height , and correlation time of turbulence , can be varied and can even be allowed to depend on radius .in addition to obtaining numerical solutions for our rather general set of equations , we also obtained numerical solutions from successively more idealized versions of these equations , all of which are used to some extent in the dynamo literature . by comparing such solutions we were able to assess the suitability of these approximations , at least for the parameter values considered .possibly the greatest shortcoming of both the numerical and analytical models is the neglect of the gaseous halo , which has been observed in several edge - on galaxies to emit polarized synchrotron radiation , and probably contains large - scale magnetic field .nevertheless , our model is , we believe , a solid base for exploring more realistic solutions in the future . in the meantime , it is hoped that our analytical solution provides a viable starting point for researchers who are interested in painting magnetic fields onto galaxies in order to explore various observational signatures .the author thanks k. subramanian for many useful discussions during the development of the numerical code .the author is also grateful to a. shukurov for providing comments on an early draft , and to the anonymous referee for feedback that significantly improved the presentation .the toroidal and poloidal parts of equation  and equation  can respectively be written as  \biggr.\\ &    \biggl.-\eta\left[\frac{{\partial}\psi}{{\partial}r}\frac{{\partial}t}{{\partial}r }          -t\lambda^-\psi + \frac{{\partial}\psi}{{\partial}z}\frac{{\partial}t}{{\partial}z}\right]\biggr\}\\ &    -\frac{1}{r}\frac{{\partial}}{{\partial}r}(r{{{u}}_r}\alpha{_\mathrm{m } } ) -\frac{{\partial}}{{\partial}z}({{{u}}_z}\alpha{_\mathrm{m}})\\ &    + \kappa\lambda^+\alpha{_\mathrm{m}}+\frac{{\partial}\kappa}{{\partial}z}\frac{{\partial}\alpha{_\mathrm{m}}}{{\partial}z } + \frac{{\partial}\kappa}{{\partial}r}\frac{{\partial}\alpha{_\mathrm{m}}}{{\partial}r } ,  \end{split}\ ] ] where and . in the present work ,we have adopted .this allows us to write these equations in the form , , and .

Abstract: we present an effectively global analytical asymptotic galactic dynamo solution for the regular magnetic field of an axisymmetric thin disc in the saturated state . this solution is constructed by combining two well - known types of local galactic dynamo solution , parameterized by the disc radius . namely , the critical ( zero growth ) solution obtained by treating the dynamo equation as a perturbed diffusion equation is normalized using a non - linear solution that makes use of the ` no- ' approximation and the dynamical -quenching non - linearity . this overall solution is found to be reasonably accurate when compared with detailed numerical solutions . it is thus potentially useful as a tool for predicting observational signatures of magnetic fields of galaxies . in particular , such solutions could be painted onto galaxies in cosmological simulations to enable the construction of synthetic polarized synchrotron and faraday rotation measure ( rm ) datasets . further , we explore the properties of our numerical solutions , and their dependence on certain parameter values . we illustrate and assess the degree to which numerical solutions based on various levels of approximation , common in the dynamo literature , agree with one another .   [ firstpage ]  magnetic fields dynamo galaxies : magnetic fields galaxies : spiral galaxies : ism mhd


coordination of multi - agent networks has attracted a significant amount of attention in the past few years , due to its broad applications in various fields of science including physics , engineering , biology , ecology and social science . distributed control using neighboring information flow has been shown to ensure collective tasks such as formation , flocking , rendezvous , and aggregation .central to multi - agent coordination study is the study of consensus , or state agreement , which requires that all agents achieve the desired relative position and the same velocity .consensus seeking is extensively studied in the literature for both continuous - time and discrete - time models .recently also asynchronous event - triggered sampling for such consensus - seeking multi - agent system was studied .researchers are not only concerned with what connectivity conditions can guarantee consensus , but also with the convergence rate : how fast the network reaches a consensus under certain connectivity assumptions .consensus algorithms are usually carried out over an underlying communication network .thus , connectivity of this communication graph plays a key role in consensus analysis .various connectivity conditions have been used to describe frequently switching topologies in different cases .the joint connection " , i.e. , the union of graphs over a time interval , and similar concepts are important in the analysis of consensus stability with time - dependent topology .uniformly joint connectedness , which requests the joint connection is connected for all intervals which are longer than some positive constant , has been employed for different consensus problems from discrete - time to continuous - time agent dynamics , from directed to undirected interconnection topologies . studied the distributed asynchronous iterations , while proved the consensus of a simplified vicsek model .furthermore , and investigated the jointly - connected coordination for second - order agent dynamics .a nonlinear continuous - time model was discussed in with directed communications , in which convergence to a consensus is shown to be uniform within bounded initial conditions .on the other hand , -joint connection requires the joint connection is connected for infinitely many disjoint intervals in  ] .let initial condition be with .it is not hard to see that .hence , grc can not be achieved since can be arbitrarily large .we first establish the following lemma indicating that the dini derivative of is bounded above by , and the dini derivative of is bounded below by .[ lem2]for all , we have   _ proof ._ we prove .the other part can be proved similarly .let represent the set containing all the agents that reach the maximum in the definition of at time , i.e. , . then according to lemma [ lem1 ], we obtain \leq \max_{i\in\mathcal{i}(t)}w_i(t)\leq |w(t)| , \nonumber\end{aligned}\ ] ] which completes the proof .we next establish two lemmas on uqsc graphs .[ lem3 ] suppose is uqsc. then there exists a center from which there is a path for all in with , and each arc of exists in a time interval with length at least during .  _ proof ._ denote as the first moment when the interaction topology switches within ( to suppose there are switchings is without loss of generality ) .if , then , there exists a center from which there is a path for all in since is quasi - strongly connected , and moreover , each arc of path stays there for at least the dwell time during due to the definition of .  on the other hand , if , we have .then , for any , there is also a center from which there is a path for all in , each arc of which exists for at least during .this completes the proof .  suppose that is uqsc .define a set - valued function , where represents the ( power ) set containing all the subsets of :   [ lem4]assume that is uqsc and let be the ( generalized ) diameter of .then for any , there exists such that for as many as at least during  ] for all .then , the total number of the elements of all the preimages of on interval  ] since for all .then we get the contradiction and the conclusion is proved .we are now in a position to prove the sufficiency statement in theorem [ thm1 ] .assume that the initial time is for simplicity .the analysis of will be carried out on time intervals  ] .based on lemma [ lem2 ] , we see that for all  ] .assume that   _ step 1 ._ according to proposition [ lem4 ] , we can choose be the center of joint graphs , , m=1,2,\dots , d_0 ] .  with ( [ 12 ] ), we have \end{aligned}\ ] ] where , which implies (\hbar(sk_0)+k_0\| w \|_\infty  ) + e^{-\int_{sk_0}^t\mathcal{y}_{k_0}(\tau)d\tau } x_{k_0}(sk_0)+ k_0\|w\|_\infty\nonumber\\  & \leq \xi_0\ell(sk_0)+(1-\xi_0)\hbar(sk_0)+2k_0\| w \|_\infty , \;\ \ t\in[sk_0,(s+1)k_0]\end{aligned}\ ] ] where with denoting the smallest integer which is no smaller than . here the first inequality of ( [ 8 ] ) follows from gronwall s inequality , and the second one holds based on assumption a2 , ( [ 4 ] ) and the simple fact that ._ since is a center in , we can well define a set for . in this step, we will establish an upper bound for .we have for , where . using gronwall s inequalitywe thus obtain where the second inequality follows from the facts that and furthermore , noticing that we conclude from ( [ v1 ] ) that where .therefore , ( [ 6 ] ) implies that for any , there exists an instance such that applying inequality ( [ 7 ] ) on for  ], there will be two cases .* when , one has , furthermore , , such that * when , one has , then , such that .taking , we obtain hence the sufficient part is proved .( necessity . )suppose is not uqsc .then , , such that .moreover , , such that is not quasi - strongly connected .furthermore , we define and in the same way as the proof of theorem [ thm1 ] .let initial condition be with and .then it is not hard to find that .therefore , the global asymptotic consensus can not be achieved since can be arbitrarily large .    theorem 2 only states the sufficiency of uqsc graph for girc .let us see the following simple example which shows that the corresponding necessity claim does not hold .* example 1 * suppose there are only two nodes , and , in the network .the arc set is if for , and otherwise .take .then we have   it is not hard to see that the system in example 1 achieves a girc , though the varying communication graph is not uqsc .therefore , uqsc graph is no longer necessary to ensure girc with general directed communications .the proof follows the same line as the proof of theorem [ thm1 ] .we will bound on time intervals  ] , we have , \;i=1,\dots , n.\ ] ]  suppose is a node as defined in the proof of theorem [ thm1 ] .provided , without loss of generality , that and as that \end{aligned}\ ] ] which implies (\hbar(sk_0)+\eta_s  ) + e^{-\int_{sk_0}^t\mathcal{y}_{k_0}(\tau)d\tau } x_{k_0}(sk_0)+ \int_{sk_0}^{t } e^{-\int_{sk_0}^z\mathcal{y}_{k_0}(\tau)d\tau } |w(z)|dz\nonumber\\  & \leq \xi_0\ell(sk_0)+(1-\xi_0)\hbar(sk_0)+2\eta_s , \;\ \t\in[sk_0,(s+1)k_0]\end{aligned}\ ] ] where the second inequality follows from the simple fact that .therefore , similar to the proof of theorem [ thm1 ] , the analysis can be carried on node by node for different disjoint intervals and we can eventually arrive at consequently , for any , it holds that thus , together with the observation that the following girc inequality is obtained : this completes the proof .  a sharper inequality can be obtained based on the first inequality of ( [ v2 ] ) that which will be useful in the following discussions on distributed event - triggered consensus .this section focuses on the proof of theorem [ thm3 ] . inwhat follows of this section , we assume that the communications over the network is bidirectional , i.e. , is bidirectional graph for all .we introduce a partition , , for the time - axis .let .then can be defined by induction as see that when is ijc , is finite for any fixed .we can further define then characterizes how many times for different proper joint graphs being jointly connected during time interval .the necessity part of the conclusion is straightforward , so we just focus on the sufficient part .denote . then based on lemma [ lem2 ], we have for all .we divide the following analysis into five steps ._ step 1 . _ take with . denote by since is infinitely jointly connected , we have that according to the definition of .note that , we have because no other node is connected to during  ] , where ._ denote is a neighbor of in \big)\big\} ] .  based on similar analysis with ( [ 6 ] ), we can conclude from ( [ 91 ] ) that where ._ in this step , let us further discuss the bound of after for nodes in .let us view as a switching instance in the graph signal for any .denote as the first switching instance after .. there will be two cases :  * if , we can relax ( [ 91 ] ) and ( [ 92 ] ) on time interval  ] .otherwise , equals the first time instance when ( [ t0 ] ) holds .if the triggering condition is totally determined by equation ( [ t0 ] ) , a node will never be triggered again once its control input equals zero at some time .consequently , a global consensus can not be guaranteed .this is why the forceful waking up ( timeout ) condition is introduced to the triggering condition .next , the communication and updating protocol for the considered distributed event - triggered coordination control is stated as follows .  * _ ( broadcasting ) _ each agent broadcasts its state during until it is triggered another time at . *_ ( receiving ) _ agent can receive if and only if there exists a time such that is a neighbor of at time .moreover , agent can store this message until another message from is received . * _ ( updating ) _ each agent updates its control input at time once it is triggered , based on the messages it receives from the neighbor set .  sticking to rule( i)-(iii ) , we present the control rule for each node : where with .  denote , and then we can write ( [ t1 ] ) into the following form :   furthermore , denote for and as the difference between two consecutive event time instances , and let be their lower bound .note that , if , the behavior , which indicates infinite triggering in finite time , is then avoided .take as the case with in the definition of in ( [ rate ] ) .denote the main result on distributed event - triggered coordination is stated as follows .suppose with and .let satisfy < \frac{1}{2}.\ ] ] then system ( [ 0 ] ) with control law ( [ t1 ] ) achieves a gac with if is uqsc ._ define a function as the lower bound for the inter - event times before time . is obviously non - increasing . moreover , for any , can be triggered at most times during because .thus , based on the definition of , we see that then ( [ v6 ] ) implies \delta ( t),\ \ i=1,\dots , n.\end{aligned}\ ] ]  noting that , for any , each arc in will still be kept in the communication graph defined by neighbor sets  .hence the dwell time assumption still stands , and therefore , we can conclude from ( [ v3 ] ) that \delta(\tau)d \tau\nonumber\\ & \leq a_0 e^{-\theta_0t}\mathcal { h}(x^0)+2(n-1)(4d_0 + 1)\big [ 1+\frac{e^{2\theta l_0 } l_0}{m(t)}\big]a_0 ^ 2e^{-\theta_0t}\int_0^t e^{\theta_0\tau}\delta(\tau)d \tau\nonumber\\ & = a_0 e^{-\theta_0t}\mathcal { h}(x^0)+2(n-1)(4d_0 + 1)\big [ 1+\frac{e^{2\theta l_0 } l_0}{m(t)}\big]\frac{a_0 ^ 2a}{\theta_0-\theta}\big(e^{-\theta t}-e^{-\theta_0t}\big)\end{aligned}\ ] ]  _ claim . _ .we prove the claim by contradiction .assume that .then we have . without loss of generality ,let us assume that for all and . from ( [ tr1 ] ) , we have \delta ( t_k^i).\ ] ] now we can further conclude from ( [ v7 ] ) that \delta ( t_k^i ) } \cdot e^{-\theta\tau^i_{k+1}}\nonumber\\ & \geq  \frac{a m(t^i_{k})}{\big[a_0 \mathcal { h}(x^0)+\frac{2(n-1)(4d_0 + 1 )  a_0 ^2 a}{\theta_0-\theta } + 2a \big]m(t^i_{k})+ 2a\big[\frac{(n-1)(4d_0 + 1 )  a_0 ^2 } { \theta_0-\theta}+1\big ] l_0 e^{2\theta l_0 }  } \nonumber\\ & \ \ \ \times \frac{e^{-\theta\tau^i_{k+1}}}{n-1 } .\nonumber\end{aligned}\ ] ]  now we know that for any fixed number , there exists such that when ,   } \cdot{e^{-\theta\tau^i_{k+1}}}.\ ] ]  since , there has to be as tends to infinity such that . on the other hand , with ( [ 99 ] ) , we can choose and sufficiently large to enforce   } \cdot e^{-\theta\tau^{i_0}_{k_0 + 1}}>1.\ ] ] as a result , ( [ 100 ] ) will lead to which contradicts the fact that is non - increasing .the claim is proved .  with , we further obtain \delta ( t),\end{aligned}\ ] ] which guarantees gac for system ( [ 0 ] ) immediately according to proposition [ pro0 ] .the desired conclusion follows .this paper focused on the robustness of continuous - time consensus algorithms of single integrator .we provided a precise answer to how much connectivity is required for the network to agree asymptotically based on noisy communications .the idea of input - to - state stability and integral input - to - state stability inspired us to our definitions of robust consensus and integral robust consensus .we showed that uniform joint connectivity is critical with respect to robust consensus for general directed graphs ; infinite joint connectivity is critical with respect to integral robust consensus .upper bounds for the -convergence time were obtained as a straightforward result of the robustness analysis .the results may have many applications since the assumptions we use are quite general . by a generalized integral version of the weight assumption, the dynamics can cover many models in both theoretical and application study . as an illustration , we studied distributed event - triggered coordination using the robust consensus inequality .u. mnz , a. papachristodoulou , and f. allgwer . robust consensus controller design for nonlinear relative degree two multi - agent systems with communication constraints ._ ieee transactions on automatic control _ , vol .56 , no . 1 , 145 - 151 .

Abstract: the paper investigates consensus problem for continuous - time multi - agent systems with time - varying communication graphs subject to process noises . borrowing the ideas from input - to - state stability ( iss ) and integral input - to - state stability ( iiss ) , robust consensus and integral robust consensus are defined with respect to and norms of the disturbance functions , respectively . sufficient and/or necessary connectivity conditions are obtained for the system to reach robust consensus or integral robust consensus , which answer the question : how much communication capacity is required for a multi - agent network to converge despite certain amount of disturbance . the -convergence time is then obtained for the network as a special case of the robustness analysis . the results are based on quite general assumptions on switching graph , weights rule and noise regularity . in addition , as an illustration of the applicability of the results , distributed event - triggered coordination is studied .   * keywords : * multi - agent systems , robust consensus , joint connection , convergence rate , event - triggered coordination


the mobile data traffic has been doubling each year during the last few years and the wireless industry is preparing a -fold increase in data demands expected in this decade . to deal with this challenge , the fifth generation ( 5 g ) communications system has come at the forefront of wireless communications theoretical research .two main approaches in 5 g are massive antennas and dense deployments of access points , which lead to the massive multiple - input multiple - output ( mimo ) and small cell techniques .massive mimo employs hundreds of antenna elements at the base station ( bs ) to serve tens of users simultaneously at the same time - frequency resource .the large size of transmit antenna array not only significantly increases the capacity through excessive spatial dimensions , but also averages out the effect of fast channel fading and provides extremely sharp beamforming concentrated into small areas . aside from these, the huge degrees - of - freedom offered by massive mimo also reduce the transmit power .nevertheless , due to the finite channel coherence time , performance of massive mimo is mainly limited by pilot contamination which arises from pilot reuse among adjacent cells . on the other track ,small cell improves the system capacity by densely deploying low - power access points into the traditional high - power macro cells . in this fashion ,distance between transmitter and receiver can be significantly reduced which results in remarkably enhanced rate gains . as small cells do not always have direct links to the macro bs , they can be intelligently deployed in accordance to the traffic demand without much cost on the fiber usage and real estate . however , the performance of small cell is mainly affected by the additional inter - cell interference induced from massive transmitting nodes .  as both massive mimo and small cells have attractive attribute in capacity enhancement , it s natural to wonder which one performs better under which scenarios .comparison of massive mimo and small cell from special and energy efficiency is addressed in . however , model the small cell system as a one - tier network with dense bss , which is unreasonable since small cell is designed to offload heavy traffic from macro bs . in , a two - tier architecture including massive - antenna macro bs and small - cell access points is explored , but it only considers the single - cell scenario and ignores the randomness of bss locations .the flexible add - on small cells and further influence on user associations make it necessary to capture bss locations into analysis .therefore , a reasonable framework to compare massive mimo and small cell should be a heterogeneous cellular network ( hcn ) containing different types of multi - antenna and randomly located bss .stochastic geometry provides a useful tool to describe spatial distribution of bs sites and traffic flow .based on this , lots of work have focused on the hcn with poisson point process ( ppp ) distributed bss . a general multi - tier framework is proposed in , and the coverage probability with flexible biased cell association is analyzed in .on top of these , further extensions to multi - antenna transmission are presented in , which is much more challenging than the single - antenna scenario . however, none of these works can be applied directly on massive mimo systems , since they do not take into account channel estimation and the further effect of pilot contamination , which is the main limiting factor of massive mimo . moreover ,all these works adopt the single - slope path loss model with only non - line - of - sight ( nlos ) transmissions , which is not fit for dense small cell networks since the short propagation distance results in more line - of - sight ( los ) transmissions and multi - slope path loss affects the benefit of densification remarkably .  in this paper , we propose a framework for downlink hcn where the user location as well as the deployment of multi - antenna macro and small bss are modeled as independent ppps . in this network , users are flexibly associated with the strongest bs , and each bs simultaneously serves multiple users associated with it on the same time - frequency resource block .further , the signal propagation experiences a path loss model differentiating los and nlos transmissions , and the channel state information at bss is acquired through uplink training . using stochastic geometry tools, we quantify the rate performance in a general setting that accounts for the interference affecting both the channel estimation and data transmission phases .our main contributions are summarized as follows :  * we propose a general framework for the analysis of downlink hcn which consists of ppp distributed macro and small cell bss with multiple antennas .our analysis captures the essential keypoints of both massive mimo and small cells , including los / nlos transmission , bs deployment density , imperfect channel estimation , and random network topology . * based on our analytical results , we compare the system performance between densifying small cells and expanding macro bs antennas .it is found that adding small cells into the network can improve the achievable rate much faster than expanding antenna arrays at the macro bs .however , when the small cell density exceeds a critical threshold , the spacial densification will stop benefiting and further impair the network capacity .in contrast , the achievable rate always increases with growing antenna size and saturates to an upper bound caused by pilot contamination .this upper bound is also larger than the peak rate obtained from deployment of small cells .* we provide the optimal small cell density that maximizes the achievable downlink rate , which can be used as a rule - of - thumb for practical small cell deployment . to fully exploit the degrees - of - freedom offered by the large antenna array ,the optimal bias with different antenna numbers is provided as guidance for practical massive mimo configuration .* we also investigate the effect of distributed and centralized antennas .it is found that to attain higher data rate with fixed budget of antenna number , taking certain amount of antennas into distributed small cell bss is more beneficial than centralizing all antennas at the macro bs .the optimal antenna allocation proportion is also presented as a reference for practical antenna configuration while combining with the hardware constraint .the remainder of this paper is organized as follows .section [ sec : system model ] introduces the hcn framework , los / nlos transmission model , and the user association policy . in section[ sec : achievable downlink rate ] , we derive a tight approximation of the achievable downlink rate accounting for channel estimation with uplink training . in section [ sec : numerical results ] , we provide numerical results to validate the analytical results and further study the performance of the hcn .finally , section [ sec : conclusion ] summarizes the main results of this paper ._ notation_throughout the paper , vectors are expressed in lowercase boldface letters while matrices are denoted by uppercase boldface letters .we use to denote the conjugate - transpose of , and use {ij} ] , with  ] is the channel vector from to the mbs , and represents the additive white gaussian noise ( awgn ) matrix with i.i.d .zero - mean elements and variance .similarly , the noisy pilot matrix at scb can be written as where ^t ] . therefore , we get that . similarly , the precoding vector for scb is given as where .  to this end , if the typical user is associated with mbs , the downlink sinr is given by where if the typical user is associated with the scb , denoted as , the downlink sinr is given by where    based on the sinr given in and , the achievable downlink rate of the typical user can be expressed as where , , , and denote the received sinr of the typical user when it is associated with the mbs in los path , nlos path , and with the scb in los path and nlos path , respectively . to facilitate the rate derivation, we make the following assumptions .  [ assumption 1 ] we approximate the coverage region of the mbs as a ball centered at with radius  .let be the point process formed by locations of the user in each macro cell .note that is a perturbation of the process and thus not a ppp .obtaining the exact correlation between and requires complicated mathematical derivations and is highly intractable . therefore, we use the similar method in to model the interfering users in , denoted by , as an inhomogeneous ppp .motivated by the fact that the probability that a user scheduled by the mbs in a los path is , and in a nlos path is , where is the distance of this user to , we make the following assumption .[ assumption 2 ] the point process can be approximated as an inhomogeneous ppp with density being .\ ] ]  moreover , for , and are independent .  based on all the analysis and assumptions mentioned above, a tight approximation of the achievable downlink rate is given in the following theorem .[ theorem 3 ] the achievable downlink rate of the typical user is approximated by where \right)\notag\\ & \times \exp{\negthickspace}\left({\negthickspace}-2 \pi \lambda_{\tt s } \left[\int_{x_3}^\infty { \negthickspace}\left(1-e^{-\frac{z p_{\tt s}l^{\tt l}}{\rho_1 u^{\alpha^{\tt l } } } } \right){\mathbb{p}}_{\tt s}^{\tt l}(u ) u du + { \negthickspace}\int_{x_4}^\infty{\negthickspace}\left(1-e^{-\frac{z p_{\tt s}l^{\tt nl } } { \rho_1 u^{\alpha^{\tt nl } } } } \right)\left(1-{\mathbb{p}}_{\tt s}^{\tt l}(u)\right ) u du \right]\right),\end{aligned}\ ] ] and \right)-\exp{\negthickspace}\left(-\frac{z x_1}{\rho_1}\left[x_5+\frac{x_3 x_5 ^ 2}{x_2\left(x_5 { \negthickspace}+{\negthickspace}x_4{\negthickspace}+{\negthickspace}\sigma^2/\tau p_p\right)}\right]\right),\end{aligned}\ ] ] while du ] , , , , where with ] , as well as , , , and .  see appendix [ proof of theorem 3 ] .  from theorem [theorem 3 ] , we can observe the effects of and on the achievable downlink rate as described in following remarks . notethat increasing means deploying more small cells , and increasing means expanding antenna array at the mbs .[ remark 2 ] it s not easy to determine the behavior of with respect to directly , but we can facilitate it with the aid of as well as numerical results . from , we know that the downlink rate in is a monotonic function of . meanwhile , fig .[ fig changewithmm ] reveals that the downlink rate grows as the increment of .therefore , it can be deduced that the downlink rate monotonically increases with at all the feasible region , which also coincides with our general intuition .when , the saturation of the downlink rate to a constant value can be directly observed form .[ remark 3 ] when increases , more users will be associated with scbs , so and grow , and and reduce , which further result in the impact of on downlink rate decays .larger can shorten the transmission distance and improve the achievable rate , but when exceeds a critical threshold , the aggravation of interferences due to switches from nlos to los will instead impair the rate performance .equation quantifies how all the key features of an hcn , i.e. , los / nlos transmissions , interference , and deployment strategy affect the achievable downlink rate .the validation of analysis as well as several numerical results based on will be shown in section [ sec : numerical results ] to give more practical insights into the design of hcn .in this section , we validate the accuracy of our analysis through simulations and evaluate the performance of the hcn via numerical results . in particular , we compare the performance of downlink rate achieved by massive mimo and small cells .then , the optimal network configuration parameters are provided as useful guidance for practical implementations .finally , we explore the antenna allocation between centralized mbss and distributed scbs .  from a practical perspective, we use the linear los probability function adopted in 3gpp for both the mbs and scb path losses , which is given by according to , parameters used in our simulation are set as follows : km , , , , , dbm , dbm , and dbm .the uplink pilot power is dbm , and the mbs density is bss / km .   in fig .[ fig compare_assoprob ] , the simulated user association probability is compared with our analytical results in theorem [ theorem 1 ] . clearly , we can see perfect agreement between the simulated and analytical values , which justifies the accuracy of our calculations .when increases , as expected , the probability of user association with the mbs reduces and that with the scb grows .moreover , all the probabilities change with almost linearly , which coincides with the conclusion in remark [ remark 1 ] that the average number of users associated with each scb remains unchanged as increases .[ fig compare_assoprob ] also reveals that the macro cell users are more likely in nlos transmissions while small cell uses are more likely in los transmissions .this is because the sparse deployment of mbss leads to a large serving distance between users and their associated mbss , which results in rare los paths , and in contrary , the dense scbs make the los connection to a user more often .  , where .,width=332,height=294 ]  in fig .[ fig compare_withm ] , the simulated achievable downlink rate is compared with the analytical approximation in under different values of and .we can see that the analytical approximation and simulation results fairly well match and follow the same trend , thus verifies the accuracy of theorem [ theorem 3 ] .due to the tightness between the simulations and analysis , we will use the latter for our following investigations . note that the number of bs antennas should be larger than the number of users it serves .hence , is set to be larger than , and is set according to the average number of users associated with each scb as given in corollary [ corollary 1 ] . from fig .[ fig compare_assoprob ] , we can get the curve slop by line fitting and obtain that .therefore , the average number of users associated with each scb is .  , and .,width=332,height=294 ]   in this subsection , we aim to compare the performance of massive mimo and small cells .in particular , fig .[ fig changewithlambdascb ] reveals the effect of increasing with fixed , while fig .[ fig changewithmm ] shows the effect of increasing with fixed . for fairness, we set the same starting configuration for these two cases , where the mbs antenna array is and the scb density is bss / km .  fig .[ fig changewithlambdascb ] shows that the downlink rate burgeons with the increment of until reaching a critical threshold , after which the expectancy for further improvement breaks into a slow decline . this is because when is small , the network can benefit a lot from the small cell densification due to the reduced distance between transceivers .however , when becomes large , more and more interference paths switch from nlos to los , resulting in an aggregated interference that significantly impair the rate performance .this observation is consistent with the conclusion in where the set up is with single antenna bss .the critical scb density threshold , i.e. , the optimal that maximizes the downlink rate , is marked out by black dots .note that as the scheduled user number grows , the optimal also grows , since more small cells are needed in a more crowded environment .the optimal for different is summarized in fig .[ fig optimallambdascb ] .[ fig changewithmm ] shows that the donwlink rate increases monotonically with respect to , but it can not grow without bound and saturates to a constant value limited by pilot contamination . it can be also observed that the smaller leads to a better rate performance , because we are considering the achievable rate per user , and less number of users gives less interferences .  by comparing figs .[ fig changewithlambdascb ] and [ fig changewithmm ] , we can see that adding small cells into a sparse network is more effective in boosting up the achievable rate than expanding antenna arrays at mbss .densification is remarkably beneficial when is low , where the rate can be enhanced almost linearly .for example , when , we observe that increasing by can bring almost rate gain .that means , on average , adding small cells into per macro cell can double the downlink rate . in contrast , for the massive mimo system in fig .[ fig changewithmm ] , we need to add more than antennas to have the same rate gain . however , the peak rate obtained from small cell densification is lower than that from antenna array extension .besides , as more number of antennas can tolerate more users in a single cell , massive mimo achieves better sum rate than small cell technique .therefore , we conclude that the deploying small cells can improve the achievable rate fast and effectively , but it will stop benefiting and further impair the network capacity when the scb density exceeds a critical threshold . on the contrary , though the massive mimo technique improves the system performance slower , but expanding antenna size can always benefit the network capacity , and the maximum rate with large is greater than that obtained from small cell deployment . in summary , if the rate demand is low , deploying smalls cell is preferred due to its rapid rate gain ; but if the rate requirement is high , the massive mimo technique is more preferable due to the higher achievable rate it provides .  , where , and .,width=332,height=294 ]  , where bss / km , and .,width=332,height=294 ]   in this subsection , we are interested in obtaining the optimal scb density that can maximize the achievable downlink rate as shown in fig . [ fig changewithlambdascb ] .moreover , with abandoned antennas available at mbss , the biasing policy should be adjusted to fully exploit the excessive degrees - of - freedom .therefore , we also investigate the impact of biasing factor in this subsection .  fig .[ fig changewithlambdascb ] illustrates how the optimal varies with the number of users scheduled by per mbs , , under different bs antenna numbers .we can see that the optimal increases monotonically with , since the most effective way to offload traffic caused by crowed users is deploying more scbs .we also find that the optimal increases and decreases with , which coincides with general intuition as a powerful mbs with large antenna array can provide sufficient data rate for users in its coverage thus requires less scb deployment , while scbs with more antennas are desirable for rate enhancement thus the optimal increases .   vs. number of scheduled users per macro cell , where .,width=321,height=294 ]  fig .[ fig optimalb ] presents the optimal scb bias factor under different mbs antenna number .clearly , the optimal decreases as grows , since larger can bring more significant rate gain thus a smaller is desired to push more users associated with mbss and benefit from the vast degrees - of - freedom .we also note that the optimal also decreases with increment of due to a similar argument .all the above results can serve as useful guidance for practical network design .   vs. number of mbs antennas , where and .,width=332,height=294 ]   in this subsection , we explore the antenna resource allocation between the mbs and scbs under fixed amount of antenna budget , i.e. , we aim to investigate how many antennas should be allocated to the mbs and scbs , respectively , such that the system performance can be maximized .let be the total number of antennas per macro cell .then , on average , . the proportion of antennas assigned to scbs per macro cell is defined as .[ fig changewithratio ] shows the achievable downlink rate as a function of for a fixed .we can see that the rate first increases as grows , which indicates that spare certain amount of centralized mbs antennas to the distributed scbs can improve the system performance .however , when exceeds a critical value , the rate begins to abate , which means assigning too many antennas to scbs will leave mbs without enough spatial diversity and impair the network capacity .we also note that the optimal that maximizes the downlink rate , as marked out by black dots , increases with .  , where and .,width=332,height=294 ]  more precise information about the optimal is summarized in fig .[ fig optimalratio ] .we find that the optimal grows with scb density while remains unchanged for different .this is because more antennas need to be assigned to scbs when their density increases , while the user density has no impact on the optimal portion .particularly , the optimal is limited by an upper bounded smaller than , since we can not allocate all antennas to scbs as the mbs needs to schedule users and assign pilots before data transmission. the practical antenna allocation can get useful references from this figure while combining with hardware constraints .   vs. scb density ,where .,width=332,height=294 ]in this paper , we have developed a framework for downlink hcn that consists of randomly distributed mbss and scbs with multiple antennas , and the los and nlos transmissions are differentiated . using stochastic geometry ,we have derived a tight approximation of achievable rates to compare the performance between densifying small cells and expanding bs antenna arrays .interestingly , we have found that adding small cells into the network is more effective in boosting up the achievable rate than expanding antenna arrays at mbs .however , when the small cell density exceeds a critical threshold value , the spatial densification stops benefiting and further impairs the network capacity .in contrast , expanding bs antenna array can always improve the capacity until reaching an upper bound caused by pilot contamination , and this upper bound is larger than the peak rate obtained from deployment of small cells .therefore , for low rate requirements , the small cell is preferred due to its sheer rate gain ; but for a higher rate requirements , the massive mimo is preferred due to better achievable rate .the optimal scb density is also presented as a guidance for practical small cell deployment .moreover , we have found that allocating part of antennas to the distributed scbs is better than centralizing all antennas at the macro bs , and the optimal allocation proportion has also been provided for practical configuration reference . in conclusion , this work has provided a further understanding on how to leverage small cells and massive mimo in future hcns deployment .in general , the typical user can receive four types of transmit powers , i.e. , from the mbs through los and nlos path , and from scb through los and nlos path , respectively .the typical user is associated with the mbs in a los path means that this received power is higher than other three cases , which can be formulated from and as follows & = { \mathbb{p}}\left [ \left\ { \frac { p_{\tt m } l^{\tt l } } { \left(s_{\tt m}^{\tt l}\right)^{\alpha^{\tt l } } } > \frac{p_{\ttm } l^{\tt nl } } {  \left(s_{\tt m}^{\tt nl}\right)^{\alpha^{\tt nl } } }  \right\ } \bigcap \left\ { \frac{p_{\tt m } l^{\tt l}}{\left(s_{\tt m}^{\tt l}\right)^{\alpha^{\tt l } } }  > \frac{bp_{\tt s } l^{\tt l } } { \left(s_{\tt s}^{{\tt l}}\right)^{\alpha^{\tt l } } }  \right\ } \bigcap \left\ { \frac{p_{\tt m } l^{\tt l}}{\left(s_{\tt m}^{\tt l}\right)^{\alpha^{\tt l } } } > \frac{b p_{\tt s } l^{\tt nl } } { \left(s_{\tt s}^{{\tt nl}}\right)^{\alpha^{\tt nl } } }  \right\ }  \right]\notag\\ & = \! { \mathbb{p}}\!\left[s_{\tt m}^{\tt nl } \!>\ !l}}{\alpha^{\tt nl}}}|s_{\tt m}^{\tt l } \!=\ ! r\right ] \! \cdot \ !{ \mathbb{p}}\!\left[s_{\tt s}^{{\tt l } } \!>\ !k_2 r|s_{\tt m}^{\ttl } = r\right ] \ ! \cdot \ ! { \mathbb{p}}\!\left[s_{\tt s}^{{\tt nl } } \!>\ !r^{\frac{\alpha^{\tt l}}{\alpha^{\tt nl}}}| s_{\tt m}^{\ttl } \!=\ ! r\right ] \! \cdot \ !f_{s_{\tt m}^{\tt l } } ( r ) dr . \ ] ] with results from void probability , the first term in the integral of can be calculated as follows & =  { \mathbb{p}}\left[\text { no bs in } \phi_{\tt m}^{\tt nl } \text { is closer than } k_1r^{\frac{\alpha^{\tt l}}{\alpha^{\tt nl}}}\right]\notag\\ & = \exp\left(-\int_0^{k_1 r^{\frac{\alpha^{\tt l}}{\alpha^{\tt nl } } } } \lambda_{\tt m}(1-{\mathbb{p}}_{\tt m}^{\tt l}(u ) ) 2 \pi u du\right).\end{aligned}\ ] ] similarly , we can derive the second and third terms in respectively as follows & = \exp\left(-\int_0^{k_2 r } \lambda_{\tt s}{\mathbb{p}}_{\tt s}^{\tt l}(u ) 2 \pi u du\right),\notag\\ { \mathbb{p}}\left[s_{\tt s}^{{\tt nl } } >k_1 k_3 r^\frac{\alpha^{\tt l}}{\alpha^{\tt nl}}|s_{\tt m}^{\tt l } = r\right ] & = \exp\left(-\int_0^{k_1 k_3 r^\frac{\alpha^{\tt l}}{\alpha^{\tt nl } } } \lambda_{\tt s}(1-{\mathbb{p}}_{\tt s}^{\tt l}(u ) ) 2 \pi u du\right).\end{aligned}\ ] ] moreover , the pdf of can be obtained as )}{dr}=\exp\left(-\int_0^r \lambda_{\tt m } { \mathbb{p}}_{\tt m}^{\tt l } ( u ) 2 \pi u du \right )  { \mathbb{p}}_{\tt m}^{\tt l } ( r ) 2 \pi r \lambda_{\tt m}.\ ] ] therefore , follows from the substitution of into . by performing the same procedure ,we can obtain the probabilities  ] , and  ] .the event is equivalent to that given that the typical user is associated with the mbs in a los path , i.e. , ={\mathbb{p}}\left[s_{\tt m}^{\tt l } > r | e_{\tt m}^{\tt l}\right ] = \frac{{\mathbb{p}}\left[s_{\tt m}^{\tt l } > r , e_{\tt m}^{\tt l}\right ] } { { \mathbb{p}}\left[e_{\tt m}^{\tt l}\right]}.\ ] ] a similar argument as in leads us to the following calulation & = \int_r^\infty  { \mathbb{p}}\left[p_{\tt m } l^{\tt l } \left(s_{\tt m}^{\tt l}\right)^{-\alpha^{\tt l}}>p_{\tt m } l^{\tt nl } \left(s_{\tt m}^{\tt nl}\right)^{-\alpha^{\tt nl}}| s_{\tt m}^{\ttl } = x\right]\notag\\ & ~~~~~~\times { \mathbb{p}}\left[p_{\tt m } l^{\tt l } \left(s_{\tt m}^{\tt l}\right)^{-\alpha^{\tt l}}>p_{\tt s } l^{\tt l } \left(s_{\tt s}^{{\tt l}}\right)^{-\alpha^{\tt l}}b | s_{\tt m}^{\ttl } = x\right]\notag\\ & ~~~~~~\times { \mathbb{p}}\left[p_{\tt m } l^{\tt l } \left(s_{\tt m}^{\tt l}\right)^{-\alpha^{\tt l}}>p_{\tt s } l^{\tt nl } \left(s_{\tt s}^{{\tt nl}}\right)^{-\alpha^{\tt nl}}b| s_{\tt m}^{\tt l } = x\right]f_{s_{\tt m}^{\tt l}}(x ) dx,\end{aligned}\ ] ] by using results in , we have & = \frac{2 \pi \lambda_{\tt m}}{\mathcal{a}_{\tt m}^{\tt l } }  \int_r^\infty x { \mathbb{p}}_{\tt m}^{\tt l}(x ) \zeta_1\left(x , k_1 x^\frac{\alpha^{\tt l}}{\alpha^{\tt nl}}\right)\zeta_2\left(k_2 x , k_1 k_3 x^\frac{\alpha^{\tt l}}{\alpha^{\tt nl}}\right ) dx.\end{aligned}\ ] ] the pdf of is then follows from taking derivative of with respect to .the pdf of , and can be obtained from the same procedure , which are omitted due to space limits .we take the derivation of for example . with the maximal - ratio - transmission precoder ,when the typical user is associated with the mbs in los path , from , we can get where denotes the average over path loss , follows from the approximation in ( * ? ? ?* lemma 1 ) , and is given as next , we calculate the expectations in separately .  from and , we know the variance of elements in and are and , respectively . hence , we have moreover , where is the element of the channel estimation vector , and is obtained from the substitution of as well as some basic algebraic operations .  for , we obtain where is obtained from the independence of and , and for , we obtain which is got from the independence of and .when , the result is dependent on the association of , which leads to the following discussion :  * if , i.e. , is associated with the mbs , we have which is obtained according to the similar procedure as . for , which is obtained according to the similar procedure as , and for  , * if , i.e. , is associated with the scb , denoted as , we have for , and for ,    considering that the path loss from interfering cells are much smaller than that from the associated cell , we approximate the large - scale fading coefficients of interfering cells by their means . therefore to denote when the expectation is only averaging over the large - scale fading . ] , we have next , we try to derive the expectations in .  for ,when is associated with the mbs in los , we obtain =\mathcal{a}_{\tt m}^{\tt l},\ ] ] and similar results can be got for associated with the mbs in nlos , associated with the scb in los and nlos , respectively .the average number of association users in should be calculated conditioned on or .then , we have considering that the interference from other cells are generally quite small , according to _ assumption [ assumption 1 ] _ , we can get the following approximation from the campbell s theorem as follows : ,\ ] ] and .\ ] ] moreover , according to _ assumption [ assumption 1 ] _ and _ [ assumption 2 ] _, for any , the expectation of the path loss from all interfering users are given by du,\ ] ] and du.\ ] ] the substitution of all these expectations into gives therefore , becomes where is from the continuous mapping theorem , and ,\ ] ] ,\ ] ] while . with the probability generating functional of the ppp , we have \right),\end{aligned}\ ] ] and \right).\end{aligned}\ ] ] the rate can be obtained by substituting into .following the same procedure , we can also derive the rate , , and , respectively .then , the desired result can be obtained from .v. jungnickel , k. manolakis , w. zirwas , b. panzner , v. braun , m. lossow , m. sternad , r. apelfrojd , and t. svensson , `` the role of small cells , coordinated multipoint , and massive mimo in 5 g , '' _ ieee commun . mag ._ , vol .52 , no . 5 , pp . 4451 , may 2014 .f. rusek , d. persson , b. k. lau , e. g. larsson , t. l. 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that the total number of mbss in the network is and the total number of scbs is , we can get average number of users associated with each mbs as and the average number of users associated with each scb as

Abstract: we develop a framework for downlink heterogeneous cellular networks with line - of - sight ( los ) and non - line - of - sight ( nlos ) transmissions . using stochastic geometry , we derive tight approximation of achievable downlink rate that enables us to compare the performance between densifying small cells and expanding bs antenna arrays . interestingly , we find that adding small cells into the network improves the achievable rate much faster than expanding antenna arrays at the macro bs . however , when the small cell density exceeds a critical threshold , the spacial densification will lose its benefits and further impair the network capacity . to this end , we present the optimal small cell density that maximizes the rate as practical deployment guidance . in contrast , expanding macro bs antenna array can always benefit the capacity until an upper bound caused by pilot contamination , and this bound also surpasses the peak rate obtained from deployment of small cells . furthermore , we find that allocating part of antennas to distributed small cell bss works better than centralizing all antennas at the macro bs , and the optimal allocation proportion is also given for practical configuration reference . in summary , this work provides a further understanding on how to leverage small cells and massive mimo in future heterogeneous cellular networks deployment .   heterogeneous cellular networks , los / nlos , massive mimo , small cells , stochastic geometry