the unique properties of water , namely its polar nature , high dielectric constant , and its ability to form hydrogen bonds , are responsible for the existence of life as we know it . at the same time , these very properties are the main obstacle in modelling hydration effects . an accurate model of ion and small molecule hydration will therefore have important applications in computational chemistry , chemical engineering , and drug design .  in the latter ,the interest stems from the fact that drug - like small molecules are usually moderately polar to achieve the required bioavailability . therefore their solvation free energy is generally dominated by the electrostatic contribution , while the nonpolar contribution is more or less negligible .one of the most successful models for predicting solvation free energies of small molecules ( and proteins ) therefore employs the poisson or poisson - boltzmann equations . within this approach ,the solvent is typically treated as a structureless medium of a given dielectric constant ( for water , . )the solute itself is treated as a low- `` cavity '' .clearly , this approach fails to take into account many relevant details of the solvent structure .the presence of long - ranged electrostatic fields induces static correlations among the polar molecules that may vary considerably in space depending on the molecular build - up of the solute .these can be taken into account by performing molecular dynamics simulations , thus leaving the continuum description at the expense of high computational cost even for small molecules .  including more structural detail into the continuum description on the one hand without increasing computational complexity on the other is therefore highly desirable .at the same time , however , also the avoidance of additional parameter sets , which is common practice in the use of continuum electrostatics in computational chemistry , is a key requirement .  in this work ,we follow one particular route to achieve this goal .we develop an approach for the computation of the electrostatic ( polar ) contribution to the free energy of solvation based on nonlocal electrostatics .this continuum approach , originating in the physics literature as a generalization of classical electrostatics _ to account for media with spatial dispersion _ , is developed here within a physically transparent minimal parameter set and applied to the simplest systems of single ions and small molecules for which experimental and theoretical results are available for quantitative comparison .in particular we address the issue of a systematic choice of ion radii which most commonly is based on essentially empirical parametrizations . we conclude with an outlook on extensions of our approach for more complex , and biologically relevant , molecules .the electrostatic potential of a charged molecule in solution is , within classical ( local ) electrostatics , given by the poisson equation  = -\varrho({\bf r})\ ] ] where denotes the dielectric constant of vacuum , and is a local dielectric function which is taken as the constants of the solution ( water ) , and a much smaller value within a solute molecule ( for proteins , e.g. , a typical value taken is ) .the rhs of eq.([one ] ) is the charge density giving rise to the potential .this arguably simple description is prone to complications : first , the transition region between solute and solution is ill - described and needs a further ( rather delicate ) modeling .second , very little is said about structural effects due to the orientation of the polar water molecules near highly charged regions of a given solute molecule .  the simplest way to introduce structural effects into a continuum description of the solvent is to account for correlations due to polarization effects between solution molecules characterized by a correlation length .this gives rise to a nonlocal generalization of eq.([one ] ) given by  where isotropy of the liquid medium is supposed .eq.([two ] ) needs to be accompanied by suitable boundary conditions ( see ) .these conditions can be simplified considerably by assuming the atomic constituents as conducting spheres , which is exact for the case of ions , and approximate for the case of small molecules .the solution of the nonlocal solvation problem with a treatment of the full boundary conditions is beyond the scope of this work and will be reported elsewhere .  the rationale behind eq.([two ] )is the following .since the electrostatic fields are long - ranged , the solvent molecules will feel the presence of the fields of other molecules over characteristic distances ( that will be of particular relevance below ) . maintaining locality of the physical fields ( i.e. , electrostatic potentials and electric fields ) , it is the response functions into which this purely static correlation effect needs to be embedded .we note that the dielectric function is related to the dielectric response function via , irrespective of the local or nonlocal character of the theoretical description .  due to the assumption of spatial isotropy , eq.([two ] )lends itself to a treatment in fourier space , reducing the linear integro - differential equation eq.([two ] ) to an algebraic equation involving the fourier - transformed function with wavevector .explicit functional dependences of either or have been derived from various approaches , e.g. within a ginzburg - landau theory for the polarization fields .  motivated by these workswe have formulated a family of nonlocal dielectric functions fulfilling the following minimal requirements :  \i ) for high -values ( equivalently , on small spatial scales ) the solvent molecules can not follow the polarization forces .  thus , a limiting value for the macroscopic dielectric function has to be reached in this limit .typical choices in the literature are  , or  .we have considered both cases and do not find a significant dependence on this choice ; + ii ) in the opposite limit , , the value for should equal the macroscopic value ; + iii ) the physical length scale of the polarization fluctuations is characterized by a correlation length .the local electrostatic limit is given by ; + iv ) causality conditions ( kramers - kronig relations ) need to be fulfilled .+ while the conditions i)-iv ) are clearly not sufficient to determine the dielectric function unambiguously , we find that the following class of functions in fourier space fulfill these criteria sufficiently well \ , , \ ] ] which depend on only two parameters , , and . for ,the dielectric function decays exponentially in real - space , while is a lorentzian for , decaying in real - space as .note that formally the case was discussed previously in , based , however , on a different physical interpretation ( see also for a similar approach in the context of nuclear physics ) .we have also tested alternative choices , e.g. a gaussian model and a nonlocal model derived from a ginzburg - landau theory for polarization fields .the comparison of our nonlocal models with other approaches and experimental data is made possible by using the concept of an _ effective _ local dielectric function .it is usually defined for the potential of a point charge , where ^{\ , -1}\ , .\ ] ] for a general charge distribution can be defined as well .we note that our determination of the effective dielectric constant is consistently performed within the nonlocal continuum theory and does not rely on data - based fits .expression ( [ five ] ) allows to test our approach against previous results .we took empirical models for the radial dependence of derived for experimental results from mehler and eichele ( me ) and conway ( co ) .the co - model is reproduced best by the lorentzian ( ) for , while the me - model is reproduced best for .the purely exponential model ( ) displays unphysical singularities in at small , while with a choice of , the co - model is reproduced for larger distances , .figure 1 shows , as computed by eq.(5 ) for a single sodium ion , using the lorentzian model with .the computation of performed here makes use of a specifically adapted fft applicable also to molecules .  a gaussian choice for ,however , leads to strong oscillations in the effective dielectric function , although its overall shape resembles the co - model for .checking our approach against the nonlocal model by sutmann et al . , we find that their theory fails to comply with our requirement iii ) , i.e. , it does not reproduce the correct limiting value at large distances .we now turn to the application of our approach to small ions .we treat these ions as born spheres .a standard problem in the definition of born - type ions is the definition of the ion radius .starting from the accepted interpretation that a solvated ion is surrounded by solvation shells , the first of these shells will be `` as close as possible '' to the ion .we can therefore identify the position of the centers of the oxygen in the first solvation shell with the first peak in the radial distribution function ( rdf ) which can be obtained either from scattering experiments or from molecular dynamics simulations .we defined the radius of the water oxygen as half the position of the first peak in the oxygen - oxygen radial distribution function ( rdf ) of bulk water .the ion radii were then derived by subtracting this radius from the position of the first peak of the ion - oxygen rdf derived from a molecular dynamics simulation .  as a specific input for our calculationswe take the values for the ion radii determined by qvist , obtained from a combined free energy perturbation / force field approach based on the gromos force field , employing several different water models ( flexible spc , rigid spc , and tip3p ) .based on this input , we have determined the solvation free energy from nonlocal electrostatics , which is given by where is the dielectric flux density .the result of the calculation is shown in figs . 2 for monovalent ions and in fig . 3 for divalent ions , for all models to be compared here .note that we chose the parameter based on its fit to the effective dielectric function .the co - model was reproduced by for ( exponential model ) , by for ( lorentzian ) and by for the gaussian model .the me - model could only be reproduced by with .the value for the exponential model is in accord with the findings in ref . .also note , as shown in table 1 , that the values we obtain are only marginally corrected ( if at all ) by effects due to nonlinear saturation , which contributes in principle as well to a reduction of the dielectric constant near the ion .the model by sutmann was used with two different limiting values ( sutmann 1 with , and sutmann 2 with ) .as the figures show , our results are consistently better than all the other theoretical curves .the computation of electrostatic solvation free energies is also possible by employing eq.([five ] ) in a standard ( local ) poisson solver .we have implemented this in an available library ( ball ) .4 compares the results of our computations for the monovalent ions with the nonlocal theory based on eq.([six ] ) and from the effective local dielectric function , eq.([five ] ) , demonstrating the consistency of both approaches .5 finally compares the theoretical results of the local and nonlocal approaches , obtained with the poisson solver , with the experiment values . evidently , the nonlocal approach yields results consistently superior to those obtained from the standard local theory .  we stress the significant advantage of our computations to work with first - principle radii without _ arbitrary _ adjustments . in figs . 2 - 5 ,the qvist - radii were used .in addition , we have also tested the shannon - radii derived from x - ray crystal data , without significant effect on our results .our result thus give a basis to the general belief that the need to introduce effective radii is _ in fact _ a consequence of the local water structure around the ions .we believe that our nonlocal approach demonstrates that it is therefore preferable conceptually to introduce the length - scale governing the structural effects in the solvent rather than introducing necessarily artificial procedures to adjust the born radii .as a second application we apply our approach to determine the solvation free energy for small alcohols for which we can assume that the polar contribution exceeds the nonpolar one . fortunately , accurate measurements of the solvation free energies of these molecules are available .we write the charge distribution of these molecules as a linear superposition of radially symmetric partial distributions translated by a vector , , i.e. .its fourier transform is then given by the expression   again we have to define the radii of the atoms .for this we chose to classify the atoms into classes depending on their chemical environment in the molecule ( e.g. , like the hydrogen atom in an oh - group ) , expecting that the radii of all atoms in a certain class are more or less similar . for methanol ( ) , e.g. , we defined four classes of atoms and used their rdf s with the oxygen of water to find the following set of radii : methyl c : 2.135 , hydroxyl o : 2.014 , hydroxyl h : 1.115 , methyl h : 1.394 .compared to the calculation of the free energies for the ions this computation is slightly more involved as it requires a three - dimensional integration instead of a one - dimensional one . within the framework of the simplified boundary conditions , the unperturbed fields of the atomic constituents are simply superimposed . to ensure short computation times , in our implementation we have used the vegas - monte - carlo integration scheme that is supplied with the gnu scientific library gsl .details of these computations will again be given elsewhere .the results for some small alcohols are shown in table 1 . for the comparison we have used the exponential with  , the lorentzian with  , and the lorentzian with  .all values in these tables are given in kj / mol .the interpretation of the results is more complicated than in the born ion case . while there we could assume that the contribution of the nonpolar part of the free energy of solvation could be neglected ( for the born ions it can be estimated to be of the order of kj / mol ) , this is not the case for the alcohols .the electrostatic contribution is still the dominant part , but not the only significant one .we therefore applied a very simple model for the nonpolar contribution in order to be able to compare our results to experimental data .the results given for the polar contribution can also be optimized by considering improved charge distributions .we have demonstrated that the approach to the computation of solvation free energies based on nonlocal electrostatics is able to reproduce experimental data for ions and small molecules with reasonable accuracy .the approach we put forward has the basic advantage to rely on essentially one parameter which has a transparent physical interpretation as the correlation length of the polarization fluctuations . while within the present paper this parameter is used as a fitting parameter , it should be clear that it might be also determined by experiment or simulation .a further significant advantage is that our approach does not make use of the commonly used adjustments of atomic or ionic radii . a challenge for the future will be the extension of our approach to determine electrostatic solvation free energies for more complex and biologically relevant molecules .work in this direction is in progress . + * acknowledgement .* this work is supported by the dfg under schwerpunktsprogramm informatikmethoden zur analyse und interpretation grosser genomischer datenmengen " ( grant le952/2 - 1 ) .  p. ball , _ h : a biography of water _ ,weidenfeld and nicholson ( 2000 ) b. honig , k. sharp , a .- s .yang , j. phys . chem . * 97 * , 1101 ( 1993 ) b. honig , a. nicholis , science * 268 * , 1144 ( 1995 ) r.m .jackson , m.j.e .sternberg , j. mol .* 250 * , 258 ( 1995 ) r.m .levy , e. gallicchio , annu .chem . * 49 * , 531 ( 1998 ) j.d .jackson , _ classical electrodynamics _ , john wiley & sons , inc .( 1998 ) r.r .dogonadze , a.a .kornyshev , a.m. kuznetsov , teor . mat .fiz . * 15 * , 127 ( 1973 ) a.a .kornyshev , a.g .volkov , j. electroanal .chem . * 180 * , 363 ( 1984 ) a.a .kornyshev , in _ the chemical physics of solvation _ , r.g .dagodnaze et al ., elsevier , amsterdam ( 1985 ) g. sutmann , _ die nichtlokale dielektrische funktion von wasser _, phd - thesis , jlich ( 1999 ) m.v .basilevsky , d.f .parsons , j. chem .phys . * 108 * , 9107 ( 1998 ) m.v .basilevsky , d.f .parsons , j. chem .phys . * 108 * , 9114 ( 1998 ) m.s .lee , f.r .salsbury jr . , c.l .brooks iii , j. chem .phys . * 116 * , 10606 ( 2002 ) a. hildebrandt , o. kohlbacher , r. blossey , h.p .lenhof , in preparation ( 2003 ) a. hildebrandt , _ an algorithm for the efficient and reliable computation of the electrostatic contribution to the free energy of solvation using nonlocal electrodynamics _ ,diploma thesis , saarland university ( 2002 ) ( unpublished ) j.b .hasted , _ aequous dielectrics _ ,chapman & hall , london ( 1973 ) l.d .landau , e.m .lifschitz , _ electrodynamics of continuous media_ , pergamon oxford ( 1960 ) we have normalized the fourier transform symmetrically .u. ritschel , l. wilets , j.j .rehr , m. grabiak , j. phys .g : nucl . part .18 * , 1889 ( 1992 ) e.l .mehler , g. eichele , biochemistry * 23 * , 3887 ( 1984 ) b. mallik , a. masunov , t. lazaridis , j. comp . chem . * 23 * , 1090 ( 2002 ) b.e .conway , j.o.m .bockris , i.a .ammar , trans .faraday soc .* 47 * , 756 ( 1951 ) j. qvist , j.phys. chem . * 94 * , 8021 ( 1990 ) w.f .van gunsteren and h.j.c .berendsen , groningen molecular simulation ( gromos ) library manual version 3.0 o. kohlbacher , h .-lenhof , bioinformatics * 16 * , 815 ( 2000 ) l. sandberg , o. edholm , j. chem . phys . * 116 * , 2936 ( 2002 ) r.d .shannon , c.t .prewitt , acta crystallogr . , sect .b : struct . crystallogr .chem . * 25 * , 925 ( 1969 ) r.d .shannon , acta crystallogr . sect .a : cryst .. crsytallogr . * 32 * , 751 ( 1976 ) g.p .lepage , j. comp . phys .* 25 * , 192 ( 1978 ) to be found at : http://sources.redhat.com/gsl h. uhlig , j. phys41 * , 1215 ( 1937 ) y. marcus , ion properties , dekker , new york ( 1997 )  . comparison of different models for the hydration free energy of different mono- and divalent cations .all energies are in kj / mol . is the hydration free energy computed for a born ion of radius . is the electrostatic hydration free energy computed with our nonlocal lorentzian model ( ) . is the nonpolar contribution to the hydration free energy as computed with the model of uhlig . is the nonlinear correction to the hydration free energy for the corresponding ion ( taken from ) . is the experimental hydration free energy ( taken from  ) . . . [ cols="^,^,^,^,^,^,^,^,^",options="header " , ]

Abstract: solvation free energy is an important quantity in computational chemistry with a variety of applications , especially in drug discovery and design . the accurate prediction of solvation free energies of small molecules in water is still a largely unsolved problem , which is mainly due to the complex nature of the water - solute interactions . in this letter we develop a scheme for the determination of the electrostatic contribution to the solvation free energy of charged molecules based on nonlocal electrostatics involving a minimal parameter set which in particular allows to introduce atomic radii in a consistent way . we test our approach on simple ions and small molecules for which both experimental results and other theoretical descriptions are available for quantitative comparison . we conclude that our approach is both physically transparent and quantitatively reliable .   solvation free energy , nonlocal electrostatics


more than forty years ago , kolmogorov and chaitin laid the foundations of algorithmic information theory , by introducing the concept of algorithmic information content , or kolmogorov complexity , for a given string of information .this measure of complexity is defined as the length of the shortest possible program on a universal computer that will output the string in question . herewe propose a conceptually analogous measure of the complexity of any connected physical structure . instead of a universal computer which translates a program into a string of information , we consider a general framework of self - assembly rules , which act together to create a physical object .the ` program ' now is our set of self - assembly building blocks and rules , the ` computer ' is given by the physical interactions of the self - assembling building blocks , and the ` output ' is the final structure . using this approach we investigate the physical complexity of shapes in two and three dimensions , including polyominoes , molecules and protein complexes .our work generalizes ideas first explored in , and opens them up to a wide range of applications .furthermore , in the context of protein complexes it offers the kind of biological application of information - theoretic concepts demanded in .there are many examples of self - assembling structures in physics , chemistry and biology .examples include thin films , micelles , viruses and dna .our aim is to introduce a general framework for the theoretical study of self - assembling structures .this framework can be used to study the properties of real self - assembling systems , but , more generally , it can also be used to measure the physical complexity of any construct , self - assembling or not .the exact nature of the self - assembly framework depends on the underlying physical system , but it always contains two basic ingredients : a set of building blocks and a set of rules .we shall call this combination an _ assembly kit _ .each building block has interfaces , which typically are subject to geometric constraints ( depending on the physical system ) . attached to each interface of a given building block is an integer $ ] .the possible values of these integers are the _ colours _ of these interfaces .the number of distinct colourings of the building blocks depends entirely on the geometry of the problem .the second ingredient of the assembly kit is the set of rules , which takes the form of an interaction matrix between colours . in the simplest casethis matrix is binary , where 1 signifies attraction and 0 signifies no interaction at all .many more sophisticated interaction matrices involving repulsion and a continuous spectrum of energies are easily imaginable .  for any system of self - assembling particleswe need to also specify a model for the actual assembly process .a convenient choice is a model assuming a single nucleus in solution , which makes the assumption that each disjoint object has one fixed nucleus building block which is surrounded by a solution containing a freely moving population containing many copies of each type of building block .each time step ( i ) a fixed building block , ( ii ) a site adjacent site to it , ( iii ) a random rotational orientation , and ( iv ) a building block from the solution are chosen at random , and the new , randomly rotated building block becomes fixed to its position if the rules allow it . note that some assembly kits always assemble into the same shape - these we call ` deterministic ' - while ones which contain ambiguous rules are ` non - deterministic ' .see figure [ crossblob ] for an example of a deterministic and a non - deterministic self - assembly kit .  as a simple example of a self - assembling system, we will consider self - assembling _polyominoes_. a polyomino ( also known as a _ lattice animal _ ) is a set of connected sites on a ( typically square ) lattice .these connected sites are our self - assembly building blocks .every building block has four sides ( so that for all ) , which are painted with one of colours .these colours can attract each other or not , as encoded in a binary interaction matrix .each distinct way of colouring a building block corresponds to a different building block _ type_. we do not regard rotated colourings as distinct .the geometry of the 2d lattice gives rise to a particular set of building block colourings in the context of self - assembly .if we have colours , the total number of such colourings is : these particular colourings are also known as _ necklaces _ , which can be defined as equivalence classes of strings under rotation .the definition of necklaces used here assumes that the building blocks have a fixed chirality - in other words that the necklaces which the colours form on the building blocks are _fixed_. necklaces . in generalwe will assume fixed chirality . ]every deterministic assembly kit , which always assembles into a structure , requires a certain amount of information to describe it in some given language .our aim is to minimize this quantity , as we define the length of the description of the _ minimum assembly kit _ as the _ complexity _ of structure : in analogy to the concept of kolmogorov complexity .any symmetry or modularity which the structure contains decreases the amount of information required to describe the structure and will therefore be reflected in its minimum assembly kit , and by extension in the value of .if a minimum assembly kit is deterministic , an interaction matrix ( with elements ) between a total of colours , of which self - interact , can be rewritten as : \delta_{i(j+1 ) } + ( i \ , { \rm mod } \ , 2 ) \delta_{i(j-1)}\ ] ] for , and otherwise , so that one colour always only interacts with one other colour . with this constraint ,the amount of information , in bits , required to describe a self - assembly kit , with building block types , is : the first term relates to the number of self - interacting colours , the second measures the information required to describe which colours out of the total of colours appear on building block , and the third term measures the information describing the distinct arrangement of the colours on the faces of building block . for a general building block with labelled faces , takes the form of : where , and the signify the number of times colour occurs on block .  for polyominoes , where is the number of necklaces with_ exactly _ colours , given by with .it follows that , , and . as before ,the complexity of polyomino is the minimum of over all possible assembly kits .note that wang tiles are a special case of self - assembling polyominoes .the tile system described in is also similar to our framework for the case of polyominos , but ( like wang tiles ) only considers self - interacting colours , and treats rotated tiles as distinct . as a result our encoding , based on necklaces , makes symmetry and modularity in the structure more directly measurable .if the faces are geometrically unconstrained - as one would imagine for a node with a set of freely moving links - and hence unlabelled , we would only need to specify how much there is of each colour .this can be written using , so that .however , this only works under the condition that multiple connections between the same pair of building blocks are prohibited .the general algorithm we use to find the minimum assembly kit , and thus the complexity , for polyominoes and other structures is described in the following section .below we describe a general algorithm for minimizing the assembly kit size for a connected physical structure without relying on steric effects . taking these into account can minimize the assembly kit even further , but their computation is highly dependent on the geometry of the system and in most cases non - trivial ( see discussion ) .note also that in some structures , such as polyominoes , some edges of the contact graph can be redundant in the context of the assembly process . whether contact graph edges in general can be redundant or not depends on the nature of the structure and the assumptions connected to the self - assembly of that structure ( see discussion ) .similarly , when interfaces are defined by geometry , as for the four sides of a polyomino building block , it makes sense to introduce a neutral colour ( below ) . in systems with a varying number of interfaces on the building blocks ,neutral colours are usually not required ( ) .  to minimize the assembly kit we take the following steps :  1 .divide the structure into building blocks ( usually a natural division ) .the number of building blocks is the _ size _ of the structure , denoted .2 .  determine the equivalence of these units in terms of any additional criteria ( e.g. types of atoms , proteins ) .this categorization is the of building block .3 .  establish a contact graph for the units ( in some cases , such as molecules , this may require setting a distance cutoff ) .if edges can be redundant : _ consider the space of all spanning subgraphs of this graph .5 .  for the contact graph ( in the case of no redundant edges ) or each subgraph ( if redundant edges exist ) : 1 .classify the ( sub)graph according to the number of connections and ( depending on the geometry ) the arrangement of connections .2 .  label all nodes which are not yet labelled and which have exactly one unlabelled node among their neighbours .the new labels distinguish nodes according to their species as well as the topologically distinct label distributions among their neighbours.[rep ] 3 .  repeat step [ rep ] until all nodes are labelled or no more nodes can be labelled .all labelled nodes we define as _ category 1 _ nodes and any remaining unlabelled nodes ( i.e. nodes with at least two unlabelled neighbours ) are defined as _ category 2 _ nodes .label all category 2 nodes simultaneously according to their neighbourhoods.[rep2 ] 6 .repeat step [ rep2 ] , using the previous labellings to distinguish neighbourhoods , until labellings are stable .these final labels , for nodes in both categories , denote the building block _ types_. the number of final labels , or types , is .these can be subdivided in to category 1 building block types and category 2 building block types .the category 2 type of block is denoted degree of each building block type in the contact graph ( or subgraph ) is the number of its interfaces .the total number of colours , including neutral colours , is .the sum expression gives the number of different types of interfaces which occur between category 2 building block types .the number of colours on building block is equal to the number of building block types in its contact graph neighbour set .10 . using , , and in equation ( [ info ] ) , calculate the information required to specify this assembly kit , and thus the complexity of the structure.[finish ] 6 ._ if edges can be redundant : _ minimize this quantity over all spanning subgraphs .figure [ algo ] illustrates the crucial steps [ rep ] to [ finish ] for a polyomino .figure [ minimumkit ] illustrates how the complexity value reflects symmetry and modularity present in the structure .[ 0.24 ]  [ 0.5 ]the self - assembly approach can be used to calculate complexity values for any physical structure . in order to demonstrate the broad range of potential applications we determine the complexity of ( a ) molecules and ( b ) protein complexes .the problem of molecular complexity has been studied extensively over the past seventy years , starting with work by plya and rashevsky among others , and culminating in a seminal paper by bertz .these approaches are based on shannon entropy rather than algorithmic information theory and focus on symmetries rather than the more general concept of modularity . in molecules , we take atoms to be the building blocks and chemical bonds to be their interfaces .simple molecules , such as those in figure [ nitro ] , for which we are only interested in the bond connectivity , are an example of a structure in which none of the edges can be regarded as redundant .this is because , unlike for polyominoes , we are not assuming any inherent geometry for the building blocks . if two atoms play the same self - assembly role but represent atoms of different atomic species , they must be differentiated .this also goes for atoms connected by different bond types .for example , in glutamine ( see figure [ nitro ] ) , the oxygen atom connected with a double bond is a leaf of the self - assembly tree just like any of the ( implicit ) hydrogen atoms , but it requires a separate building block .the two molecules in our example of figure [ nitro ] are the amino acid glutamine and the explosive nitroglycerine , which both consist of 20 atoms .nitroglycerine however exhibits a much higher degree of modularity , with its three groups , and therefore has a much lower complexity of bits than the glutamine , for which the value is bits .note that nitroglycerine does not exhibit simple three - fold symmetry , but a more subtle , hierarchical modularity .such structural features would be harder to discover using traditional approaches to the measurement of molecular complexity , which do not take a self - assembly perspective and rely on shannon entropy rather than kolmogorov complexity as a measure of complexity .[ 0.7 ]  many important biochemical structures are _ protein complexes _ , consisting of several individually formed and folded protein _ subunits _ bound together to produce functional cellular machinery .these subunits may include different types of protein and several copies of the same protein .the physical structure of protein complexes , as with protein themselves , is important in determining the functionality of the complex .the manner in which the subunits bond to form the final complex is known as the _ quaternary structure _ of the complex .the 3dcomplex database contains a description of the quaternary structures of thousands of protein complexes , in terms of subunit type and inter - subunit bonding .if we have two proteins which play the same role in the self - assembling structure but are different proteins , we can choose to count them as two different building blocks ( analogous to the aforementioned distinction between atomic species in molecules ) .in the following analyses we are only interested in the connectivity of proteins ( equivalent to the qs topology level in the 3dcomplex database ) , and therefore do not distinguish between different proteins .the two protein complexes in our example of figure [ proteincomplex ] are a chaperonin complex ( _ e . coli _ chaperonin groel ; pdb identifier : 1oel ) and an allergen complex ( _ p .allergen phl p 6 ; pdb identifier : 1nlx ) .both consist of 14 proteins , but the former displays a much higher degree of symmetry and a much lower complexity value of bits , versus bits for the allergen ( which is still somewhat modular ) .more complex protein structures require more unique inter - subunit bonds types , compared to less complex structures which can re - use bonds and be constructed through simple repetition of subunits . as an increase in bond types corresponds biologically to the presence of more unique bonding sites on subunit proteins , more complex protein structurescan be thought of as requiring more evolutionary innovation to produce and would therefore be expected to occur less frequently in biological organisms .this hypothesis is confirmed by figure [ iaincomplex ] , which shows a histogram of complexity values normalized by the size of the protein complex , to avoid size effects for the 15733 protein complexes in the 3dcomplex database .the distribution closely ( ) follows a power - law decay .  [0.65 ]   correlation with a power law decay .note that in this case we do not distinguish between different types of subunit.,width=302 ]  in both of these cases - molecules and protein complexes - we assume geometrically unconstrained faces for the building blocks ; in other words , we use . while the chemical bonds of atoms and the interfaces of proteins are in fact usually constrained , this information is not part of the structural formula of the molecule or the contact graph of the protein complex .if this additional level of resolution is required , a more realistic self - assembly model can be constructed , based on the exact three - dimensional characteristics of the atoms or proteins , and using the term specified above .the self - assembly perspective provides an intuitive definition of the modularity of a structure : if part of the structure appears several times , it still only needs to be encoded once .this is why modularity and symmetry ( being a special case of modularity ) lead to more efficient self - assembly kits and a lower value of the complexity measure . formally we can define the modularity of a structure of size as the average number of times one of the different building block types in the minimum assembly kit is used in the structure , which is simply : we can furthermore define a _module _ formally as a connected set of building blocks which appears more than once in a given structure .note that modules can overlap : a subset of a module could form another module , appearing a different number of times than the whole module .the molecule in figure [ nitro]a illustrates such a case .  the majority of protein complexes in the 3dcomplex database show high modularity values ( figure [ modular ] ) with a common trend observable along the line , indicating many proteins consist of structures involving two copies of all constituent subunits .  to further illustrate how the complexity and the modularity measure the physical complexity of protein complexes , we consider two of the outliers in the complexity and modularity histograms , the high - complexity 1ohh ( figure [ iaincomplex ] ) and high - modularity 1b5s ( figure [ modular ] ) .1ohh consists of two copies of bovine f-atpase ( itself a protein complex ) in complex with its regulatory protein if .the regulatory protein binds simultaneously to both copies of the main complex , but slightly asymmetrically , leading to asymmetric interactions being recorded in the 3dcomplex database .this asymmetry results in extra information being required to describe the combined quaternary structure , and the observed high complexity value .1b5s is a multienzyme complex consisting of multiple copies of dihydrolipoyl acetyletransferase ( e2p) .the e2p protein has the potential to occupy quasi - equivalent positions , as seen in virus structures , and is also observed to form cubic complexes .the highly - modular , dodecahedral structure exhibited in 1b5s is an efficient way of grouping many copies of an active protein in a geometry that facilitates enzymatic activity : the large windows in the structure allow passage of the substrate and product between the inner cavity and the substrate .the structure of the protein subunits allows this structure to be realised with just one building block type , resulting in high modularity .( number of building block types ) and ( size of the complex ) .many protein complexes are highly modular , and this is true across a wide range of sizes . in this plot complexes of equal modularity lie on a diagonal line with positive gradient .the lines are shown for , 2 , and 10 ( , 0.5 , and 0.1 ) .the sizes of the circles show how many complexes lie at a given position .the insets show two examples ( with pdb identifiers 1kyo and 1b5s ) , with high and low modularities.,width=302 ]if we have two structures and with minimum assembly kits and , then the _ joint _ minimum assembly kit is the minimum kit which can assemble both structures if an appropriate subset of building blocks is chosen .the amount of information required to describe this kit is the _ joint complexity _ of and .this definition can easily be generalized to more than two structures .let us define as the subset of which forms structure , and as the subset of which forms structure ( note that e.g. is not necessarily equal to due to the colour minimization ) , so that .furthermore , let us define the _conditional _ minimum assembly kit as the set of building blocks we need in addition to in order to form structure . then we can write : where denotes the set theoretic difference operation .the definition of follows accordingly .hence we can also define a _ conditional complexity _ , which is the amount of information needed to describe the building blocks in . because the way we describe the assembly kit is additive in the number of building blocks , we can write since is the information required to describe the building blocks in . the relationship between and given by where is the total number of colours in and is the total number of colours in . because of the minimization of colours , .hence , if , then .  similarly , we can define a mutual minimum assembly kit , which corresponds to the intersection from this follows the _ mutual complexity _  in order to account for the relative sizes of the structures we compare using these measures , we can define relative versions of the above quantities .these are _ relative conditional complexity _ : and the _ relative mutual complexity _ note that the latter measure resembles the jaccard index . for an illustration of joint , mutual and conditional complexity , see figure [ joint_examples ] .  [ 0.45 ]_ steric effects _ for structures which contain loop structures formed by repeating units , it is possible to exploit steric effects in order to reduce the size of the assembly kit below the minimum size found by our algorithm ( which explicitly excluded such effects in its definition ) .an example of a steric effect would be a polyomino which is self - limiting in a deterministic way , purely because of the geometric constraints of the building blocks . as long as each distinct type of loop structure is formed by building blocks of a distinct species ( or set of species ), the amount of information required to describe this structure can be taken to be the same as that required to describe an infinite chain consisting of the same elements .a simple example is given in figure [ steric ] .the crucial assumption which has to hold for this simplification to work is that the geometry of the loop is specified by the species ( and , by extension , the geometry ) of the building block . for proteins as building blocks of protein complexes ,this is a very reasonable assumption . in the case of molecules it would furthermore be possible to simplify the self - assembly kit by introducing building blocks representing common small loop structures , such as carbon rings .  _multiple nuclei _ in principle one could consider beginning the self - assembly with multiple nuclei in place .multiple nuclei may , through steric hindrance or modular repetition , be used to achieve certain structures in a more efficient way , using fewer building blocks than a single nucleus would require .this reduction in complexity may however be countered in practical applications by the difficulty of achieving the required precise relative displacements of nucleus particles .it is because of these reasons that we have concentrated on a single nucleus model , as the positioning of multiple nuclei makes it much more difficult to construct a general measure of complexity .  within the single nucleus category, we further distinguish between structure with a _ specified nucleus block _ and those with _general nucleus blocks_. the former case encompasses those assembly kits which are guaranteed to produce a given output structure if and only if a specified block is used as the nucleus ( in other words , this block is placed on the substrate before other blocks are introduced to the system ) .general - nucleus assembly kits by contrast will form the same output structure regardless of which block is placed first .see figure [ nuclei ] for an illustration how specifying a nucleus can reduce the complexity of a assembly kit .[ 0.45 ]   and blocks can join the open edges that will form .this self - assembly kit has a complexity of bits .( bottom : ) a general nucleus system to produce the same structure , illustrating the required increase in complexity ( bits).,width=226 ]  which of these classes to employ in a study depends on the motivating context of the self - assembling system under consideration .if modelling assembly in a diffusion - dominated environment , for example , the order in which interacting particles meet can not be specified , so the general - nucleus model is more appropriate . in a controlled environment where a nucleus can be placed to initiate assembly, the single - nucleus model is applicable .the two cases correspond to different ` languages ' being used to measure complexity , and so care must be taken in comparative studies to only compare numerical complexity values from within one class ._ kolmogorov complexity _ our approach to measuring physical complexity is motivated by the concept of kolmogorov complexity .it is however important to note that while kolmogorov complexity itself is uncomputable due to the halting problem , our minimum is not .this is because the runtime of a finite computer program with finite output can be infinite , while the assembly time of a finite shape is always finite .it is possible to define the actual kolmogorov complexity of a shape , but this is uncomputable .our computable complexity measure forms a bound on this unattainable quantity , and is dependent on the way in which we encode the description of the assembly kit . it therefore is useful for the analysis , classification and comparison of physical structures , as long as we use a consistent encoding .we present a general approach for measuring the physical complexity of any connected structure , using the language of self - assembly .this approach is capable of detecting symmetry and modularity in a given structure , because these features significantly decrease the size of the required self - assembly instruction set .it therefore provides a powerful tool for automated classification and categorization of physical structures .in addition , the connection between self - assembly and complexity is an argument for the ubiquity of modular and symmetric features in biological systems : since many such systems self - assemble , evolving sets of self - assembly instructions are likely to yield symmetric and modular structures , as the instructions for these are more efficient to evolve .

Abstract: we present a quantitative measure of physical complexity , based on the amount of information required to build a given physical structure through self - assembly . our procedure can be adapted to any given geometry , and thus to any given type of physical system . we illustrate our approach using self - assembling polyominoes , and demonstrate the breadth of its potential applications by quantifying the physical complexity of molecules and protein complexes . this measure is particularly well suited for the detection of symmetry and modularity in the underlying structure , and allows for a quantitative definition of structural modularity . furthermore we use our approach to show that symmetric and modular structures are favoured in biological self - assembly , for example of protein complexes . lastly , we also introduce the notions of joint , mutual and conditional complexity , which provide a useful distance measure between physical structures .


quantum key distribution ( qkd ) has been the first practical realization of communication based on the laws of the quantum mechanics . as bennett andbrassard showed with their pioneer protocol bb84 , nonorthogonal quantum states , together with an unjammable classical channel , can be used to prevent an eavesdropper ( eve ) from gaining information on the key without being revealed .there are already different working realizations of bb84  .some of them  exploit entanglement in polarization to encode a random string of bits in two unbiased polarization bases of a single photon , thus following the original proposal . however , polarization is not the most suitable degree of freedom for fiber communication because birefringence usually fluctuates randomly in a fiber , making the encoded information impossible to decode .the most appealing setup for real telecom communications is based on the phase degree of freedom .  as an example we consider the setup shown in the upper part of fig.[bb84 ] , that has been used for a qkd up to 30 km  . in this arrangement a relative phase between two time - bins of a photon state is realized by the sender ( alice ) with an unbalanced interferometer and a phase modulator .the receiver ( bob ) measures incoming bins by means of a second interferometer , matched to the sender s one , set to . with a probability of 50% the phase difference between alice andbob s interferometers will be or , and their measures will be correlated ; in the other half of cases measures will be discarded by the two users .schematics of two interferometers for bb84 .the upper one , in blue , has been used for qkd up to 30 km ; the lower one , in red , is the plug - and - play setup used for qkd up to 67 km . ]another realization of phase encoded qkd , namely the plug - and - play " setup  , is reported in the lower part of fig.[bb84 ] , and it has been used for qkd up to a distance of 67 km  . using an intense pulse ,bob populates two time - bins with relative phase , and sends them to alice .alice applies a relative phase , reflects the pulses on a faraday mirror , let them pass through an attenuator , and sends them back to bob .the two pulses retrace in their backward path all the fluctuations they suffered in the forward path , thus arriving at bob s interferometer ready for being interferometrically revealed . as in the previous scheme bob sets at random , andmeasures the outcomes , completing the _qkd run pertaining to bb84 .this mechanism of automatic compensation of noise earned this setup the name of plug - and - play " .recently bostrm and felbinger  presented a variant for qkd , and named it ping - pong " cryptography ( pp ) for its peculiarity of a forward and backward use of the quantum channel .this peculiarity seems to suit perfectly with the forward and backward dynamics typical of plug - and - play setups , as the one described above , and this motivated the present work .the main advantage of pp is the deterministic nature of its _ encoding - decoding _ procedure ; its main flow is that it has been proved to be not completely secure  .a number of variants , still based on the polarization degree of freedom of a photon , have been proposed to solve this problem , some making use of entanglement  , and others making no use of it  .we envisage here a protocol with phase - encoded information to have a secure , completely deterministic , quantum communication .a few variants and applications of the protocol are also discussed .the receiving user , bob , prepares a photon in a superposition of two time - bins with a relative phase randomly chosen between the values .he sends the photon to alice , the transmitting user , who chooses one of two possible tasks , _ control mode _ ( cm ) or _ message mode _ ( mm ) , with probability and respectively ; the former realizes a control on the security of the channel , the latter the deterministic communication between the users .alice encodes a bit of information with an unitary operation on the two time - bins : she can either apply the identity operation , choosing and encoding a ` 0 ' , or introduce a ` phase - flip ' , by applying a phase between the time - bins and encoding a ` 1 ' .the photon is sent back to bob who measures it with his apparatus set with the same phase he prepared initially .in this way bob s measurement is _ deterministic _ , because the initial phase of the state ( that he does know ) is changed by an amount of 0 or ; he can thus guess alice s operation without any needs of a classical channel .alice detects the photon with her interferometer randomly set to or .after that alice prepares a _ new photon state _ with a phase , and sends it back to bob who , analogously to what he did in mm , measures it with the same phase he prepared initially .we notice that if phase difference between bob s interferometer and alice s one is or then the two users share correlated information , while in the other two cases they do not .it is straightforward to realize that alice procedure in preparing the new photon let the two users interferometers to be necessarily correlated either in the forward path or in the backward one .this entails that the cm of our protocol results as deterministic as the mm is , thus achieving that doubling of the whole rate transmission we mentioned before .none of the qubits , neither destined to mm nor to cm , is discarded .given un unjammable public channel two users can exchange information in a secure manner by means of the above protocol .eve can try to gain information by inquiring the phase of the photon both on the forward and in the backward path ; alternatively , she can prevent bob from gaining information ( dos attack  ) by randomly measuring the state of the traveling photon ; finally , on a lossy channel , she can conceal her presence behind losses  .nevertheless no attack can remain undetected by the control procedure since it is akin to a bb84 check test , performed either on the forward path or on the backward one .this guarantees the protocol is unconditionally secure , as bb84 is , even if the security threshold may be different .our protocol gives a probability of 25% to detect dos attack as well as a general attack providing eve with full information .the practical implementation requires to gather the two interferometers of fig.[bb84 ] .the resulting scheme is shown in fig.[pppp84 ] : the upper interferometer is devoted to cm , the lower to mm , and they are connected by a 1x2 fiber - coupler .the signal prepared by bob goes at random into cm or into mm with probability and , respectively .later on , when the photon is on the backward path , this device redirects it to bob .   final setup for deterministic plug - and - play cryptography .it merges the two interferometers of fig.1 , reported in blue ( cm ) and red ( mm ) . the black represents the common elements between the two , while the green indicates new components .an automatic switch between mm and cm is performed by a 1x2 coupler . ]the mm - interferometer resembles the one used for testing bb84 up to 67 km  .it is a typical plug - and - play scheme , with the unique difference that the attenuator is inserted just after the laser - diode ( ld ) , at bob s side .a laser pulse with a mean photon number much lower than unity goes through bob s interferometer and comes out into two time - bins with a relative phase and opposite polarizations . in the plug - and - play setupan intense pulse is sent to alice to provide a trigger signal for alice s electronics . in our case , as the pulse is already attenuated at bob s side , a further synchronization system , such as usually used in other practical qkd setup  , is necessary .after reflection on the faraday mirror , the photon travels back to bob , retracing exactly those paths that let him compensate any undesired phase : this makes unnecessary the stabilization of the polarization . eventually , raileigh backscattering is not a problem in this case because of the low value of the average photon number from the beginning of the protocol .the cm - interferometer resembles the one used for bb84 qkd up to a distance of 30 km  . as a time - polarization division techniqueis implemented , this task is sensible to random phase changes , and needs adjustments during the protocol runs .a value of  rad / min for slow thermal drift in the interferometer was estimated , thus requiring a compensation step every  .also in this case alice must synchronize her electronics with detectors , phase - modulator , and laser - diode using a suitable synchronization system .we proposed a setup that fully exploits the potentialities of the two - way unbalanced interferometers of certain implementations of bb84 to merge the features of plug - and - play setup for bb84 and ping - pong cryptography .the scheme is completely deterministic , both in the encoding - decoding procedure and in the control one , thus achieving a doubling in the rate of information transmission .it seems to be quite practical to implement since it is based on existing working implementations .  besides , due to its deterministic nature , several other advantages are available .bases revelation on the public channel is avoided , and this feature , together with a slightly different alice s preparation procedure during the cm , leads to a qkd more secure than that achieved by means of bb84  .furthermore a ` quantum direct communication ' ( qdc ) between users is also possible  .finally , we notice that once qdc is available also the novel protocol called ` quantum dialogue '  can be achieved : if alice delivered a message to bob by means of a secure qdc , then bob can use the message just received as the key of encryption of his answer , and simply communicate it to alice on the public channel .          g. l. long and x. s. liu , phys .a * 65 * , 032302 ( 2002 ) ; f .-deng , _ et al .a * 68 * , 042317 ( 2003 ) ; i. p. degiovanni , _ et al . _ ,phys . rev .a * 69 * , 032310 ( 2004 ) ; q .- y . cai and b .- w .li , quant - ph/0311168 ( 2003 ) .

Abstract: we present a scheme for secure deterministic quantum communication without using entanglement , in a plug - and - play fashion . the protocol is completely deterministic , both in the encoding procedure and in the control one , thus doubling the communication rate with respect to other setups ; moreover , deterministic nature of transmission , apart from rendering unnecessary bases revelation on the public channel , allows the realization of protocols like ` direct communication ' and ` quantum dialogue ' . the encoding exploits the phase degree of freedom of a photon , thus paving the way to an optical fiber implementation , feasible with present day technology .


the advent of the big data revolution has opened the door to the analysis of massive datasets on all aspects of society .new technologies have made possible the access to unprecedented amount of information on human behavior generated unobtrusively whenever people interact with or through modern technologies such as cell phones , online services , mobile applications , etc .this fact is facilitating the pursuit of a computational approach to the study of problems traditionally associated with social sciences  .over the course of the last few years , it has allowed for the development of greater insights , for instance , into human mobility  , structure of online social networks  , cognitive limitations  , information diffusion and social contagion  , the importance of social groups  or even how political movements raise and develop  .the relation between physical location and social interactions can be also explored with the new available data . in general , people tend to interact and maintain relations with geographically close peers . a tendency that gets reflected in a decay of the social interaction probability with the physical distance .this effect has been observed , for example , in phone call records and in online friendships  .furthermore , it has been shown that online  social links can be inferred from user co - occurrences in space and time and , likewise , that the location of a person can be predicted from the geographic positions of his or her online friends  .some further aspects of the relation between geography and online social contacts have been studied such as the probability that a link at a given distance closes a triangle  , the connections between users in different countries  , the social interactions and mobility in emergency situations  or the overlap between users ego networks and how it decays with the distance  .multi - parametric inference methods have been applied to empirical data with the aim of predicting link presence and users locations  .these works show that the accuracy of link prediction is considerably improved by taking into account the geographical information , and that the accuracy of location prediction is enhanced when the online social links are provided .the availability of geo - localized information has also allowed for a detailed exploration of human mobility  .the length of displacements between locations was found to follow a broad distribution , well fitted by a power - law decaying function  .the asymmetry of the travels was studied by considering ellipsoidal boundaries to the average individual displacements and analyzing the scaling of the radius of gyration .memory effects in the individual displacements was also analyzed , finding that individuals home and workplace have a considerable impact on their mobility patterns .these results motivated the introduction of several mobility models with the aim of explaining the features observed in the data  . despite the supporting evidence ,most of these models lack a connection between mobility and social interactions .  in this work, we lay a bridge between these two worlds by introducing a model coupling social tie formation and spatial mobility .preceding models considering network structure and geography are uncoupled .our model simulates the movement of individuals and creates links between them when they are physically close mimicking the effect of face - to - face interactions .we study the model both numerically and analytically and confront its results with empirical data obtained from three online social networks .we show that the model generates more realistic networks than uncoupled models .we have collected data from online social networks containing both social links and information about the users physical positions .the first dataset was obtained from twitter by means of its api  .we identify over single users , who tweeted using a gps enabled mobile device during the month of august  . if those users reported various locations in different tweets , the most recent one is taken for the purpose of the study .the other two datasets contain information referring to the users location check - ins and the social networks of gowalla and brightkite  . both were location - based online social networks , in which users can check - in at their current locations and receive information about services in the area as well as about their friends positions .gowalla and brightkite are no longer active but their data is available online  .the main statistical features of our three datasets are displayed in table [ tab : datasets ] .social interactions across country borders have particular properties and are affected by political , linguistic or cultural factors .we overcome this difficulty by restricting our analysis to the networks within each country .intra - country mobility and social contacts account for the large majority of a user activity  . for simplicity , we focus on the three major countries with more than one thousand users in each of our datasets : the united states ( us ) , the united kingdom ( uk ) and germany ( de ) . we have analyzed and modeled other countries and found similar results to the ones presented in this manuscript.  .*datasets . *number of users ( nodes ) and of links of the networks obtained from the different geo - localized datasets for the united states ( us ) , the united kingdom ( uk ) and germany ( de ) . [ cols="^,^,^,^,^,^,^,^,^ " , ]   the geo - social properties of the networks generated by the tf model are shown in figure 5 for the us and in supplementary figures [ supfig_3 ] and [ supfig_4 ] for the uk and germany , respectively .additionally , we show how each of the introduced properties contributes to the total error of the model in table [ tab : partialerr ] .the model is able to reproduce the trends in the probability , the reciprocity , the social overlap and the disparity distribution with good accuracy .the difficulties encountered with the degree distribution and the clustering as a function of the distance are not unexpected since the model does not incorporate mechanisms to explicitly enhance the heterogeneity in the agents contacts nor favor any specific dependence of the clustering on the distance .   in this sectionwe explore two null models uncoupling mobility and social interactions to help us interpret the mechanisms acting in the tf model .the first null model , the spatial model ( s model ) , is based solely on the geography and consists of randomly connecting pair of users with a probability depending on the distance , but does not take network structure into account .the second null model , the linking model ( l model ) , in contrast , is based only on random linking and triadic closure , and it is equivalent to the tf model without the mobility .we consider the two uncoupled null models and compare their results with those of the tf model . in this way, we demonstrate the importance of the coupling through a realistic mobility mechanism to reproduce the empirical networks .  the spatial model ( s model ) consists of randomly connecting pair of users with a probability that decays as power - law of the distance between them ( suggested in  ) .the exponent of the power - law is fixed at following figure 2a .the results of the s model are shown in the panels of figure 2 . while it is set to match , other properties such as , , , or are not well reproduced .the s model fails to account for the high level of clustering and reciprocity in the empirical networks and for their dependence on the distance .the error of this null model is between - for the three countries , around twice the error of the tf model ( see figure 6 ) .the linking model ( l model ) is a simplified version of the tf model , without random mobility and the box size .agents move to visit their contacts with probability , whereas with probability they do not perform any action . in this version of the model, users can connect only by random connections or when two of them coincide , visiting a common friend , which leads to triadic closure .these two processes do not depend on the distances between the users .a thorough description can be obtained with a mean - field approach ( see section [ sec : mf ] ) .the results of the l model are shown in figure 2 .due to the triangle closing mechanism , this null model creates networks with a considerable level of clustering .however , it does not reproduce the distance dependencies of , , and .the error of the l model is also around twice higher than the error of the tf model ( see figure 6 ) .the geography and the structure are coupled in the tf model through the random mobility .changes in the underlying mobility mechanism affect the quality of the results .the lowest values are obtained with the power - law distribution in the jump lengths , while normal or uniformly distributed jumps yield worse results ( e.g. , for the us the tf model has lower by and than the tf - normal and the tf - uniform models , respectively , as shown in figure 6 ) .simplified models that neglect either geography or network structure perform considerably worse than the tf model in reproducing the properties of real networks .likewise , non - realistic assumptions on human mobility mechanism yield worse results than the default tf model .to conclude , the coupling of geography and structure through a realistic mobility mechanism produces networks with significantly more realistic geographic and structural properties .   for the tf model , the two null models : spatial ( s model ) or linking ( l model ) , and the tf model with normally or uniformly distributed travel distances.,width=313 ]       the results presentedso far have been obtained at the optimal values of and .the question remains , however , of how robust these results are to changes in the values of the parameters . in figure 7 , we report the effect of varying while is maintained constant in its optimal value .the linking probability loses its power - law shape for very low values of , marking the limit in which random mobility is the main mechanism for the agents traveling in detriment of friend visits . in this case ,most of the links are created due to encounters occurring in nearby locations or are random connections , and so the distribution of triangles disparity loses its bimodal shape .furthermore , the friend visits provide opportunities to reciprocate the connections .this is why for extremely low values of , the reciprocity is close to zero . towards the other limit , _i.e. _ , the social overlap and the triangle - closing probability steadily increase . in this limit , the linking probability , the reciprocity and the distribution of triangles disparity recuperate their shapes of the optimum .  figure 8 explores the impact of varying while is fixed to its optimal value .the effect of on and is the opposite to that of : these metrics decrease at all distances with increasing .the reason for this is that visits to friends are the main forces behind the creation of new triads and the subsequent closure of triangles .note that the more connections are created randomly ( higher ) , the less links will be a result of friend visits .we will expose and describe in detail the interplay between these two mechanisms in the mean - field calculations .  a possible variation of the tf model consists of eliminating friend visits or random connections ( _ i.e. _ , setting or to ) .this prevents the model from producing networks with characteristics comparable to the real ones in all the cases , leading to increase in of around .interestingly , the model results are quite robust to variations in the update rules , the random connection mechanism , the connecting rules in each agent neighborhood and the variants in the way users visit friends .these variations lead to changes in smaller than .a detailed discussion of the results with different model variants is included in appendix b.   in this section , we consider the l model , introduced in section [ sec : insights ] , to gain some analytical insights on the mechanisms ruling the final network structure .although this model is a simplified version of the tf model , the results of the simulations yield a relatively low value of ( figures 6 , and supplementary figures [ supfig_9 ] and [ supfig_10 ] of appendix b ) .we write the equations for the time evolution of the properties of the network and solve them numerically . among all the properties , we focus on the average clustering coefficient , the overall reciprocity and the degree distribution .the clustering coefficient is defined as a ratio of all the closed triads to all triads existing in the network , _i.e. _ , .the number of triads can be calculated knowing the degree distribution .the number of closed triads in the l model grows with time mostly due to the friend visits mechanism .a triangle is formed every time two friends of the same hosting agent meet in the host s place and decide to connect .note that an undirected triangle corresponds to undirected closed triads .assuming that the contribution of random links is negligible , the evolution of the number of closed triads is described by where , meaning that we do not distinguish between in - degree and out - degree ; represents the number of nodes with the degree higher than , _i.e. _ , the number of potential hosts , is an estimate of the lower bound for the number of triangles closed by one closing link .finally , is the expected number of encounters per host , which can be calculated as where is the number of nodes with a given degree in the network .finally , note that the above definition of degree and the one obtained from symmetrizing directed networks ( used in previous sections ) are related by a proportionality factor .the reciprocity of connections can be expressed as , where is the number of reciprocated links , is the number of non - reciprocated links and the total number of links .the numbers of links evolve as where corresponds to the probability that an agent visiting a neighbor gets her connection reciprocated ( their connection is initially single directional ) .as can be seen , , and are mutually dependent .to calculate the degree distribution , we estimate the probability of a node to increase its degree by one unit in the current time step due to multiple encounters with friends of her friends where . in the l model , however , every node can increase its degree by multiple links in each time step . for simplicity, we neglect higher order terms induced by the possibility of creating multiple links .moreover , we note that equation ( [ pk ] ) is a good estimate if there is not a strong correlation between node degrees .the number of nodes of certain degree is given by where is an estimate of the probability that the node degree increases , is the number of nodes with 0 out - degree and non - zero in - degree .such nodes are important because their connection can be easily reciprocated as a result of a friend visit .however , these nodes are not counted directly into , and so a correction is needed to account for them explicitly , as in equation ( [ eq_degdistr3 ] ) .the number of such nodes can be calculated as   the numerical solution of this set of equations describing the evolution of the l model is shown in figure 9 .the equations accurately predict the dynamics of the clustering coefficient , the reciprocity and the degree distribution for certain values of the parameters ( _ i.e. _ , for medium and high values of , as in the lower plots of figure 9 ) .the approximation yields slightly worse results when the number of random connections is small in comparison with the number of connections created due to friend visits ( _ i.e. _ , for low values of , as in the upper plots of figure 9b ) . in the latter case ,neither the degree distribution is well approximated , probably due to the degree - degree correlations introduced through the friend visit mechanism .the mean - field analysis of the l model shows that the friend visiting mechanism is a direct cause of triangle closure and link reciprocity .equation [ eq_triangles ] , which estimates the growth of the number of triangles in the network , accounts only for the friend visiting mechanism ; yet it approximates closely the value of the clustering coefficient , also when , which controls the mechanism of random connections , is high .similarly , equation [ eq_links_p ] , which estimates the growth of the number of reciprocated connections , accounts for the friend visiting mechanism and approximates well the value of reciprocity .we introduce a model that couples human mobility and link creation in social networks .the aim is to characterize the relation between network topology and geography observed in empirical online networks .the model has two free parameters and but , despite its simplicity , it is able to reproduce a good number of geo - social features observed in real data at a country level . comparing the tf model with simplified null models , we find that the coupling of geography and structure through a realistic mobility mechanism produces significantly more realistic social networks than the uncoupled models .social links in our model are formed mostly with relational ( due to triadic closure ) , and proximity ( through spatio - temporal coincidences ) mechanisms  .visiting friends helps to reinforce the existing relations and favors the closure of triads with particular properties regarding the distance balance of their edges .random link creation accounts for online acquaintances or for historical face - to - face encounters as individuals move their residence from one city to another .finally , individual random mobility allows the agents to explore new locations .an agent jumps to a new location , as opposed to slower growth reported in .we leave the exploration of temporal aspects of our model for the future research . ]our results show that by establishing an appropriate balance between friend visits and random link creation , the model can reproduce the main features of online social networks , e.g. , we show that of the mobility has to be directed towards existing friends .we demonstrate that these are the fundamental mechanisms at play in the model .the tf model is generic and functional for different datasets .human mobility driven by social ties has impact on the modeling of disease spreading , and may improve its predictions .this model can also be used in simulations of processes that involve social networks and geography , e.g. , simulations of opinion formation , language evolution , or responses of a population to extreme events . moreover, it can also be helpful to design network benchmarks with realistic geo - social properties to test , for instance , the scalability of technical solutions in social online networks related to geography of its physical infrastructure .the data analyzed are publicly available as they come from public online social sites or data repositories ( twitter , gowalla and brightkite ) .since our analysis relies on statistical features and not on single cases , any private information about users had been removed and the analysis was performed on anonymized datasets .we would like to warmly thank luis f. lafuerza for helpful discussions on the analytical treatment of the model .in this appendix , we present supplementary figures that reproduce the figures of the main text for the united kingdom and germany .in this appendix , we consider several variants of the tf model and the l model and evaluate their results .we describe a total of variants marked with different colors in the tables in supplementary figures [ supfig_9 ] and [ supfig_10 ] . for each variantwe explore the space of the parameters and .we run the models for from the set and from , yielding in total parameter combinations . for each of the model variants , we find the parameters that minimize the fitting error .we plot its value in supplementary figures [ supfig_9 ] and [ supfig_10 ] . in the following paragraphs, we describe in detail each of the variants and its results .first , we modify the jump size distribution to understand its impact on the geo - social properties .we consider the following cases : the default power - law jumps with exponent , a minimal jump length of km and a cutoff at km , as in  ( the tf model ) , uniformly distributed random jumps up to km ( tf - uniform ) , normally distributed jumps with standard deviation of km ( tf - normal ) , and no jumps to new locations at all ( the l model ) .we plot the minimal fitting error of these cases in supplementary figures [ supfig_9 ] and [ supfig_10 ] using different colors of the curves .the default power - law jumps show the best results with the lowest error for most of the variants .the normal distribution and the l model tend to perform considerably worse .the highly unrealistic uniform jumps understandingly provide the worst results and the highest error values for almost all variants .  to assess the role of friend visits and random connections we turn on and off these two components by setting to zero the corresponding parameters and .we plot the results in supplementary figures [ supfig_9 ] and [ supfig_10 ] with dashed and dotted lines .we observe significantly higher error values whenever one of these two components is turned off , for most of the model s variants , what demonstrates their importance for the tf model .  to prevent users from spreading into inhabited regions , we include in the tf model an angular preference for the jumps .namely , the direction of each jump is chosen randomly with a probability proportional to the number of inhabitants present at the destination .note that this does not affect in any way the length of the jumps , which is drawn independently beforehand . to estimate the population of the target area, we use the gridded population of the world . to test how this angular preference impacts the results , we consider a variation of the models without it and compare the results .the two variants are included in the lowest row of the table in supplementary figures [ supfig_9 ] and [ supfig_10 ] .they show almost no difference in the error values for the germany , although a systematic difference exists for the uk ; the error of the variant with direction preference is consistently lower in the case of that country .the presence of the sea around the uk introduces a distorting factor for the tf model . without the directional preference the agents freely spread over the sea independently on the geographical shape of the country , leading to unrealistic results .agents traveling and link creation can be realized in the simulation in various update orders . by default , at each time step , each agent first moves , next connects to other random agents , and then connects locally ; the following agent performs the same actions in the same order , etc .we call this method sequential ( `` seq '' ) . in an alternative update rule , which we call simultaneous ( `` sim '' ) ,first all the agents move , then all of them create random connections , and finally all of them create connections locally .the two update rules are included in the second row , counting from the bottom , of the table in supplementary figures [ supfig_9 ] and [ supfig_10 ] .the update rules have little impact on the final networks resulting from the simulation .  in the tf and l models ,the agents create with probability random connections .these links can be created in different ways ; we consider three variants .first , each agent chooses another agent uniformly at random , what constitutes the default mechanism ( `` rnd '' ) .second , each agent randomly picks another node with probability proportional to the current degree of the node , which corresponds to the preferential attachment mechanism ( `` pa '' ) .third , the agent draws another node with probability decaying as a power - law of the distance between the two agents ( `` dist '' ) , with its exponent equal to and the minimal distance of km .the type of random connecting mechanism used is listed in the third row , counting from the bottom , of the table in supplementary figures [ supfig_9 ] and [ supfig_10 ] . in some cases ,e.g. , for germany , the distant - dependent probability of creating a random link provides better results than the other variants .we consider similar variants for the connections formed inside spatial boxes , which are created with the probability .the agents can connect uniformly at random ( `` rnd '' ) , with a preference for high - degree nodes ( `` pa '' ) , or a preference toward the nodes with high intrinsic fitness ( `` fitn '' ) .the fitness of the nodes is drawn from a power - law distribution with an exponent of , which roughly corresponds to the distribution of the growth rates reported in .these variants are implemented in the following way .first , we note that the number of connections created by the agent is a result of a binomial process with probability and the number of trials is equal to the number of agents that currently stay in the given spatial box .the expected number of links created in such binomial process is known , therefore , an equivalent number of connections can be created with one of the two mentioned preferential processes .the type of connecting mechanism applied in the spatial boxes is listed in the top row of the table in supplementary figures [ supfig_9 ] and [ supfig_10 ] .there is no consistent difference in the error values between these variants .thus , the connecting mechanism applied in the spatial boxes has little impact on the results .we conclude that the main components of the tf models are crucial to reproduce the structure and geography of the social networks .these components include the mobility model , friend visits and random connections .the power - law mobility model tends to produce the best results .the angular preference of travels is important for countries whose geography is strongly restrained , e.g. , by sea . other modifications to the modelhave low or no consistent impact on the results , with the exception of the distance dependent random connections , which in certain cases consistently influence the results .leskovec j , backstrom l , kleinberg j ( 2009 ) meme - tracking and the dynamics of the news cycle . proceedings of the 15th acm sigkdd international conference on knowledge discovery and data mining - kdd 09 , p. 497506 .grabowicz pa , aiello lm , eguluz vm , jaimes a ( 2013 ) distinguishing topical and social groups based on common identity and bond theory .proceedings of the sixth acm international conference on web search and data mining - wsdm 13 , p. 627636 .borge - holthoefer j , rivero a , garca in , cauh e , ferrer a , et al .( 2011 ) structural and dynamical patterns on online social networks : the spanish may 15th movement as a case study .plos one 6 : e23883 .backstrom l , sun e , marlow c ( 2010 ) find me if you can : improving geographical prediction with social and spatial proximity .proceedings of the 19th international conference on world wide web - www 10 , p. 6170 .scellato s , noulas a , lambiotte r , mascolo c ( 2011 ) socio - spatial properties of online location - based social networks .proceedings of the 5th international aaai conference on weblogs and social media - icwsm11 , p. 329336 .volkovich y , scellato s , laniado d , mascolo c , kaltenbrunner a ( 2012 ) the length of bridge ties : structural and geographic properties of online social interactions .proceedings of the 6th international aaai conference on weblogs and social media - icwsm 12 , p. 346353 .wang d , pedreschi d , song c , giannotti f , barabasi al ( 2011 ) human mobility , social ties , and link prediction .proceedings of the 17th acm sigkdd international conference on knowledge discovery and data mining - kdd 11 , p. 11001108 .cho e , myers sa , leskovec j ( 2011 ) friendship and mobility : user movement in location - based social networks .proceedings of the 17th acm sigkdd international conference on knowledge discovery and data mining - kdd 11 , p. 10821090 .sadilek a , kautz h , bigham jp ( 2012 ) finding your friends and following them to where you are .proceedings of the fifth acm international conference on web search and data mining - wsdm 12 , p. 723732 .ratkiewicz j , conover m , meiss m , gonalves b , patil s , et al .( 2011 ) truthy : mapping the spread of astroturf in microblog streams .proceedings of the 20th international conference companion on world wide web - www 11 , p. 249252 .gridded population of the world , version 3 ( gpwv3 ) : population count grid , future estimates .palisades , ny : nasa socioeconomic data and applications center ( sedac ) .available : http://sedac.ciesin.columbia.edu/data/set/gpw-v3-population-count-future-estimates .accessed 2014 feb 6 .

Abstract: daily interactions naturally define social circles . individuals tend to be friends with the people they spend time with and they choose to spend time with their friends , inextricably entangling physical location and social relationships . as a result , it is possible to predict not only someone s location from their friends locations but also friendship from spatial and temporal co - occurrence . while several models have been developed to separately describe mobility and the evolution of social networks , there is a lack of studies coupling social interactions and mobility . in this work , we introduce a model that bridges this gap by explicitly considering the feedback of mobility on the formation of social ties . data coming from three online social networks ( twitter , gowalla and brightkite ) is used for validation . our model reproduces various topological and physical properties of the networks not captured by models uncoupling mobility and social interactions such as : i ) the total size of the connected components , ii ) the distance distribution between connected users , iii ) the dependence of the reciprocity on the distance , iv ) the variation of the social overlap and the clustering with the distance . besides numerical simulations , a mean - field approach is also used to study analytically the main statistical features of the networks generated by a simplified version of our model . the robustness of the results to changes in the model parameters is explored , finding that a balance between friend visits and long - range random connections is essential to reproduce the geographical features of the empirical networks .