Self-consistent inference formulation of gravity model

Daekyung Lee,1 Wonguk Cho,1 Gunn Kim,2 Hyeong-Chai Jeong,2 and Beom Jun Kim1, ∗

1Department of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea2Department of Physics and Astronomy, Sejong University, Seoul 05006, Republic of Korea

The gravity model has been a useful framework to describe macroscopic flow patterns in geo-graphically correlated systems. In the general framework of the gravity model, the flow betweentwo nodes decreases with distance and has been set to be proportional to the suitably defined massof each node. Despite the frequent successful applications of the gravity model and its alternatives,the existing models certainly possess a serious logical drawback from a theoretical perspective. Inparticular, the mass in the gravity model has been either assumed to be proportional to the totalin- and out-flow of the corresponding node or simply assigned the other node attribute externalto the gravity model formulation. In the present work, we propose a general novel framework inwhich the mass as well as the distance-dependent deterrence function can be computed iterativelyin a self-consistent manner within the framework only based on the flow data as input. We validateour suggested methodology in an artificial synthetic flow data to find the near-perfect agreementbetween the input information and the results from our framework. We also apply our method tothe real international trade network data and discuss implications of the results.

∗ Corresponding author: beomjun@skku.edu

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I. INTRODUCTION

The mechanism behind human mobility and flow pattern in general has long received considerable attention inbroad research areas. For example, understanding of human mobility in a city can be directly linked to the planningand optimization of various social infrastructures, and thus much interdisciplinary research has proposed a variety oftheoretical and numerical approaches[1–3].

The gravity model of trade, which is one of the most popular and representative methodologies, has provided afairly simple and general framework to handle the issue. In the scheme of the gravity model, the directed flow fijfrom the ith to jth region is usually denoted as

fij = mouti min

j Q(rij), (1)

where mouti (min

j ) corresponds to a suitably defined mass of each region, and Q(rij) is called the deterrence functiondescribing the dependency on the distance rij between i and j. Although the mass has nothing to do with the conceptof the gravitational mass in physics, the terminology has been widely accepted in the research community due to thesimilarity of the functional form in Eq. (1) to the Newton’s law of universal gravity. Despite the similarities, animportant difference exists since the directedness of the flow makes us define masses of the node in two differentdirections, inward and outward. The mass in the gravity model can be interpreted as a relative magnitude ofattraction of each region for outward and inward flows, and the deterrence function reflects that the flow is expectedto decrease with the distance between the two regions. While based on simple intuitive expectations, the gravitymodel has become an important tool to describe the trade flow[1, 4], the human migration[5–8], and also the scientificcollaboration network[9], to name a few.

In this work, we strongly claim that in spite of its wide applicability as a theoretical tool, the conventional gravitymodel faces crucial problems. First of all, the in-mass min

j and the out-mass mouti terms cannot be simply replaced

by the total inward (∑

i fij) and outward (∑

j fij) flows, respectively. The reason is because it simply violates the

scaling relation: If we double all the flows by factor c, i.e., fij → cfij , the right-hand-side of Eq. (1) becomes c2 timeslarger, leading to the clear self-contradiction. Accordingly, the proportionality constant between the total flow andthe mass should be decided carefully to remove such a problem. Also, in the other typical application of the gravitymodel, the masses have been assumed to be proportional to other quantities, without any proper justification withinthe gravity model framework. For example, the trade between two countries was simply assumed to be proportionalto the population, the GDP, or other attributes of the two countries, and precedent research focused on the findingof the proper form of the deterrence function[1, 4–8, 10]. Even though those heuristic analyses on various data haveprovided many valuable insights, the use of other attributes causes a fundamental problem in terms of the consistencyand validity of data analysis. In the present work, we demonstrate a substantially improved methodology.

Our proposed framework in this work sharply differs from existing gravity models. Only by using the flow datafij observed in reality, our framework can find all the information on the right-hand-side of Eq. (1) in a completelyself-consistent way. In other words, our methodology does not assume anything on what the masses can be, nor itassumes anything on the functional form of the deterrence function. Our novel framework uses only the flow data{fij} as input and produces mout

i , mini , and Q(rij), altogether as the outcomes from the framework, without using any

external information and assumption. We explain below our framework and test its validity first for the artificiallygenerated synthetic dataset. Our numerical results confirm that the proposed methodology can reconstruct the inputflow data with high accuracy. We then apply our proposed method to the real data of the international trade networkand discuss the implication of our findings.

II. SELF-CONSISTENT FORMULATION OF GRAVITY MODEL

In gravity models, each node corresponds to an individual region on the two-dimensional geographical surface ofthe earth, and a weighted directed link between two nodes represents the amount of a certain type of flow like humanmobility flow in the traffic network of a city and product export/import flow in a trade network. In accordance withthe spirit of the gravity model as represented in Eq. (1), the edge properties ({fij}) on the left side are assumed tobe related with the node properties ({mout

i ,min}) combined with the geographical characteristic function Q(r) on theright side. In other words, the main purpose of the gravity model is to infer the edge properties in the mobility or theflow network from the information on the node properties and a deterrence function which depends only on the nodedistance. Our proposed model to be detailed below starts from a very simple idea: For a network of the size N , thenumber of known variables in the left-hand-side of Eq. (1) is |{fij}| = N(N − 1), and in the right-hand-side we have2N unknown variables ({mout

i ,min}). The deterrence function Q(r) can also be simply replaced by a piecewise linearfunction for which M unknowns control the resolution of the approximation. Consequently, as long as the inequality

3

𝑓𝑖𝑗

𝑆𝑖out, 𝑆𝑖

in

𝑄(𝑟; 𝑛)

𝑚𝑖out(𝑛), 𝑚𝑖

in(𝑛) 𝑚𝑖

out, 𝑚𝑖in

𝑄(𝑟)

Input Iteration Output

𝑚𝑖out, 𝑚𝑖

in

𝑄(𝑟)

Ground truth

FIG. 1. The brief procedure of our Self-consistent inference scheme. As an attempt to reconstruct the input flow data fij ,we initialize the temporal mass distribution {m̃out

i (n), m̃ini (n)}, Q̄(r;n) and update them with self-consistent iterative process.

Once they converge to certain steady state, the resultant masses and deterrence function are validated with the ground truthof input data. We describe the detailed framework of our methodology in Sec. II, III.

N(N − 1) > 2N + M is satisfied for a given resolution parameter M and the network size N , it is very clear thatone can find all the information on the right-hand-side of Eq. (1) only through the use of the flows {fij} from theobservation in the real world. Of course, in doing so, the above inequality makes the problem over-determined sincethe number of knowns is often larger than unknowns for a large network, and thus we seek to minimize the differencebetween the inferred flow and the observed flow.

We propose a novel inference algorithm for the gravity model in which we can find all information of {mouti ,min},

and Q(r) in a completely self-consistent way, only by using the input information {fij} for all pairs of nodes. In ourself-consistent formulation, we first infer the mass distribution {mout

i ,min}, and combine it with the input information{fij} to infer the deterrence function Q(r) for a given resolution parameter M . The inferred deterrence function Q(r)is then used to further infer the next-stage estimation of {mout

i ,min}, and the entire process above iterates until allthe inferred information approach stationarity. The detailed procedure of our methodology consists of three steps:

(i) We start from the initial configuration at the zeroth step n = 0 in which m̄outi (n = 0) = m̄in

i (n = 0) = 1 for ∀i,and the initial deterrence function is set to Q̄(r;n = 0) = 1 at all distances r. (We have tested other initial conditionsand have found no difference in the result.) Note that we use barred symbols like m̄ and Q̄ to denote that they areinferred quantities, not observed quantities like fij .

(ii) We note that the sum over j for both sides of Eq. (1) results in∑N

j=1 fij = mouti

∑Nj=1 m

inj Q(rij), in which

the term mouti is singled out. We thus get the expression mout

i =∑

j fij/∑

j minj Q(rij), which leads to the plausible

update rule for the inference of the information at (n + 1)-th step based on information at the n-th step:

m̄outi (n + 1) =

Souti∑N

j=1 m̄inj (n)Q̄(rij ;n)

(2)

with the out-strength or the total outward flow of the ith node defined by Souti ≡

∑Nj=1 fij . It is noteworthy that all

variables in the right-hand-side of equation is either given from the real data or is known at the nth iteration step.The similar approach can be made for the summation over i for Eq. (1), which leads to the update rule for the in-massvariables:

m̄inj (n + 1) =

Sinj∑N

i=1 m̄outi (n)Q̄(rij ;n)

, (3)

with the in-strength Sinj ≡

∑Ni=1 fij . We emphasize again that the above two update rules can be performed since all

the information on right-hand-sides of equations are available from the information at the previous step.(iii) We then update the deterrence function at the (n+ 1)-th step with the updated mass distribution information

in the previous step (ii). We approximate the deterrence function as the piecewise linear function for which we need tospecify M values at Q̄(rk) with k = 1, 2, · · · ,M . For doing so, we again start from Eq. (1) and update the deterrencefunction by using

Q̄(rk;n + 1) =

⟨fij

m̄outi (n + 1)m̄in

j (n + 1)

⟩rij≈rk

, (4)

4

where 〈· · · 〉 represents the average over all (i, j) pairs with the condition rij ≈ rk satisfied. In practice, we use aspecific binning process in which the bin size is chosen in an adaptive way to include a certain fixed number of datapoints in each bin. More specifically, we sort all the data fij/m̄

outi (n + 1)m̄in

j (n + 1) in an ascending order of thedistance {rij}, and group them into M subgroups, each of which contains the same number of data points. Forthe k-th bin, we compute the average in Eq. (4) to find the value Q̄(rk) and the value of rk is set to the averageof all distances rij within this subgroup. The entire flow data ({fij}) are composed of N(N − 1) values, and wedivide them into M subgroups of equal size 500. Our binning method with equal-sized samples contributes to theenhanced accuracy of our inference algorithm since the relative statistical error for each subgroup does not changemuch, in comparison to the conventional binning method with uniform bin size for which number of data points ineach subgroup can be largely different. Note that the estimation of Q̄(rk;n + 1) in our methodology only gives Mdiscrete values. We then connect each point by linear function to construct piecewise linear continuous deterrencefunction Q̄(rij ;n) to be used in step (ii).

In summary, our self-consistent inference framework for the gravity model starts from arbitrary initial informationin (i) and first computes mass distributions in (ii), which are then used to infer the deterrence function in (iii).The whole procedure (ii) and (iii) are then iterated until all barred quantities, {m̄out

i }, {m̄ini }, and Q̄(r) converge to

stationary values. It should be noted that resulting quantities inevitably contain arbitrary scale factors: If we multiplyfactor cin for all in-mass variables, and cout for out-mass variables, and divide the deterrence function by the factorcincout, the scaling does not change the final outcomes. Only for simplicity, we fix the maximum of the deterrencefunction as unity, and at the same time the averages of in-mass and out-mass variables are set to be equal to eachother. In the following section, we validate our formalism first for synthetic data and then for the real data of theinternational trade network.

III. VALIDATION OF INFERENCE METHOD FOR SYNTHETIC DATA

In this Section, we apply our self-consistent inference formalism in Sec. II for synthetic flow data to verify itspractical applicability and to compare with existing methodologies in the conventional gravity model. In the practicalapplication, the observed flow {fij} and the distance {rij} information for all node pairs are given, and the gravitymodel in general aims to find the mass variables {m̄out

i , m̄ini } and the deterrence function Q̄(r). In this section, we

instead start from properly generated synthetic data for mass variables and the deterrence function, which we call as{m̃out

i , m̃ini } and Q̃(r), respectively. We then compute all flow information {fij} based on Eq. (1), which is then fed

into our machinery of the self-consistent formalism proposed in Sec. II, to yield our inferences {m̄outi , m̄in

i } and Q̄(r).The success of our self-consistent method can then be easily judged by comparing barred (output) and tilded (input)quantities. We display the brief scheme of our validation method in Fig. 1.

We use the international trade network composed of 111 countries as a backbone structure for synthetic data.In detail, we randomly assign all mass variables {m̃out

i , m̃ini } from the normal distribution N (1, 0.2), and set the

deterrence function as Q̃(r) = er/10000. The distance rij between a node pair in units of km is defined by the geodesicdistance, or measured along the great circle on the earth, between capitals of a given pair of countries. The flowfij computed from Eq. (1) is then used as input ground truth information for our self-consistent formulation of thegravity model.

In Fig. 2, we display how our inferred deterrence function Q̄(r;n) approaches the assumed synthetic function Q̃(r)for iteration steps n = 1, 2, and 5. Although our piecewise linear approximation combined with the specific binningmethod with group size of 500 points in Sec. II yields relatively small number of data points in Fig. 2, the consistencybetween our inferred function Q̄(r;n) and the synthetic input function Q̃(r) is remarkable even at the fifth iteration(n = 5). In Fig. 2, we also include the result of the deterrence function Q̄S(r) for the strength-based gravity modelin which mout

i ∝ Souti and min

i ∝ Sini are assumed. The estimation of Q̄S(r) for the strength-based gravity model is

made by Q̄S(r) = fij/〈Souti Sin

j 〉r=rij and we use the same binning procedure. It is shown very clearly that our self-consistent inference formalism of the gravity model can find the ground truth of synthetic network, and outperformsmuch the strength-based gravity model. Even though the result at the first iteration step is very similar to that of thestrength-based model, further iterations quickly yield the successful reconstruction of the deterrence function Q̃(r).

We next compare the inferred mass distributions from our methodology {m̄outi , m̄in

i } and the synthetic information{m̃out

i , m̃ini } in Fig. 3(a) and (b). We emphasize that our inferred mass values at the fifth iteration (n = 5) in Fig. 3(a)

and (b) show almost perfect agreement with the synthetic input data assumed as the ground truth. For comparison,we also display in Fig. 3(c) and (d) the results from the strength-based gravity model as a benchmark test. It is obviousthat the strength of each node deviates much from the input synthetic mass information, and thus we conclude thatthe strength-based gravity model fails to capture the underlying information of masses. In contrast, the inferredinformation from our self-consistent formalism shows a remarkable accordance with the underlying ground truth inthe synthetic data.

5

0.2

0.4

0.6

0.8

1

0 3000 6000 9000 12000 15000 18000

Q(r

)

r

Q‾ (r;n=1)Q‾ (r;n=2)Q‾ (r;n=5)

Q‾ S(r)Q~

(r)

FIG. 2. The inferred deterrence functions Q̄(r;n) at the n-th iteration step (n = 1, 2, and 5) for the synthetic data based on the

deterrence function Q̃(r). Clearly seen is that our self-consistent inference formalism reconstructs the deterrence function in the

synthetic data, i.e., Q̄(r;n = 5) ≈ Q̃(r). For comparison, we also include the deterrence function Q̄S(r) for the strength-basedgravity model.

IV. RESULTS FOR INTERNATIONAL TRADE NETWORK

In this Section, we apply our self-consistent inference formalism of the gravity model to the international tradenetwork. The ground truth for the real values of mass variables are not known in reality and thus we cannot compareinferred information with reality, differently from what we did in the validation in Sec. III. Instead, we apply ourformalism to infer the mass variables and the deterrence function, only based on the real observations of {fij} anddiscuss the meanings and implications of inferred information. We emphasize that our formulation of the gravity modelis sharply different from existing applications, in which the mass variables are often given from external informationbased only on simple speculations. Since the deterrence function may change depending on the choice of the massvariables, we strongly believe that existing approach has a shaky ground. In contrast, our approach in the presentwork can give both mass variables and deterrence function at the same time, without any external information otherthan the flow data {fij}.

The international trade network is a representative and very useful example data to investigate the underlyingmechanism of the global economy. Since we believe that the economic international trade relations cannot be separatedfrom other international issues like geopolitical and military relations, the trade network has become important topicin diverse research areas. In the international trade network, the node i represents a country, and a weighted directededge fij denotes the total amount of annual trade from the node i to j. We use the trade network in the year 2019to investigate the recent past.

We first perform our methodology of the self-consistent iterative inference to get the estimations {m̄outi , m̄in

i }and Q̄(r) after the sufficiently large number of iterations (n = 5). We then use the conventional strength-basedgravity model in which the in- and out-mass variables are substituted by the in- and out-strengths, respectively, i.e.,mout

i ∝ Souti and min

i ∝ Sini . The proportionality constants are fixed in such a way that the maximum of the deterrence

function is set to unity and the average in- and out-mass variables are set to equal to each other. In Fig. 4, we displaythe resulting deterrence function (a) Q̄(r) from our methodology, together with (b) Q̄S(r) based on the strength-basedapproximation of the gravity model. We first note that the deterrence functions estimated by two different methods(the conventional strength-based approximation and our self-consistent inference formalism) are not much different,and both exhibit power-law decay forms, as can easily be seen that most data points lie approximately straight linesin Fig. 4(a) and (b). However, we note that error bars are smaller and more data points appear to align along the

6

0.6

0.8

1

1.2

1.4 (a)

m−io

ut

0.6

0.8

1

1.2

1.4(b)

m −i in

20

40

60

80

100

0.6 0.8 1 1.2 1.4

(c)

Sio

ut

m~

i

out

0.6 0.8 1 1.2 1.4 20

40

60

80

100(d)

Si in

m~

i

in

FIG. 3. [(a) and (b)] Comparison of synthetic input mass information {m̃outi , m̃in

i } and the inferred information {m̄outi , m̄in

i },respectively, from our self-consistent formulation of the gravity model. [(c) and (d)] Comparison of {m̃out

i , m̃ini } with the

estimations from the simple strength-based gravity model in which in-and out-mass variables are approximated as {Souti , Sin

i }.It is seen very clearly that our self-consistent inference formulation can reconstruct the information underlying synthetic dataof {fij} while the strength-based simple gravity model fails.

0.01

0.1

1

1000 6000

(a)

Q(r

)

r

Q‾ (r)

1000 6000

(b)

r

Q‾ S(r)

FIG. 4. The deterrence functions Q̄S(r) and Q̄(r) of 2019 international trade network estimated (a) from the strength-basedapproximation of the gravity model and (b) from our self-consistent inference formalism of the gravity model, respectively.Although both deterrence functions show similar behavior of the power-law decay, our result in (a) display smaller error barsand more consistent decay form.

7

straight line in our results in Fig. 4(a), demonstrating higher consistency than the strength-based gravity model.We next report the inferred mass distributions at n = 5 in Fig. 5 in comparison to the strength values in the form of

the scatter plots in log-log scales, for (a) outward quantities m̄outi versus Sout

i and (b) inward quantities m̄ini versus Sin

i ,respectively. We first note that our inferred mass information and the strength values are highly correlated. However,detailed comparison in Fig. 5(a) appears to reveal interesting observation: Specifically, a few European countries(Germany, Poland, Slovania, etc) show relatively large values of the out-strength Sout

i with respect to correspondingvalues of the out-mass inference m̄out

i . Also, some countries in South America, Africa, and Oceania (Australia, Chile,Kenya, etc) exhibit the opposite tendency. Such deviations can also be seen in Fig. 5(b) for inward quantities.

While not definitive, we suggest that the observed behavior can be explained from the intrinsic characteristics of thegravity model. In particular, we find from Eq. (1) that

∑j fij = Sout

i = m̄outi

∑j m̄

inj Q̄(rij). Accordingly, the ratio

Souti /m̄out

i can be interpreted as the sum of inward masses of nearby nodes of j weighted by the deterrence function.Consequently, it is very plausible that the ratio Sout

i /m̄outi reflects the local economic environment of the country i: If

there are more neighbor nodes of larger inward masses around i, it tends to have larger ratio between the out-strengthand the out-mass. Our interpretation appears to be valid in Fig. 5(a); Despite having a similar out-mass value to theUS, China places further right of the US because of its multiple neighboring countries with a large economy.

Since our self-consistent inference method gives us the masses {m̄outi , m̄in

i } and the deterrence function Q̄(r), wecan then put them in the right-hand side of Eq. (1) to infer the flow distribution {f̄ij}. We next compare the flowdistribution {fij} of the real trade network and the results {f̄ij} from our self-consistent inference method in Fig. 6(a).We also compare the results for flow distribution f̄S

ij from the strength-based gravity model with the real data fijin Fig. 6(b). In Fig. 6, we exhibit the comparisons in the form of the density plots: The region with the red colorrepresents that there exist more data points in that region. In Fig. 6, we can clearly recognize that our results fromthe self-consistent inference in (a) reconstructs the ground truth values of fij much more faithfully than the strength-based gravity model in (b), although our methodology appears to overestimate the flow values for small values of fij .Hence, we conclude that the our self-consistent inference scheme is practically useful to explain the flow data in theinternational trade network.

V. SUMMARY AND DISCUSSION

In this paper, we have proposed a practically useful self-consistent inference formalism of the gravity model in whichiterative updates of mass variables and the deterrence function are made. Our method is fundamentally differentiatedfrom existing approaches in that it only uses the real flow data as input information, and identifies all informationin the gravity model in a completely self-consistent manner. Our systemic analysis of synthetic flow data has shownthat our approach successfully overcomes the limitation of prior methodologies and can reconstruct the predefinedquantities of the model network. We have also applied our methodology to the international trade network data andconfirmed that it provides a better understanding of the qualitative and quantitative characteristics of the targetnetwork.

From a theoretical perspective, we believe that our inference algorithm can be interpreted as a natural extension tothe frequently used strength approximation [10]. In the scheme of the strength approximation, an outward strengthof each node depends only on its intrinsic attributes which is proportional to the outward mass. However, in theoriginal framework of gravity model, it is actually a quantity that depends not only the outward mass of the node butalso the inward masses of nearby nodes. Our inference algorithm successfully converts the non-local information ofstrength attribute into corresponding masses, which represent purely local and intrinsic attraction of each node. Therelation between two methods also can be seen in the detailed procedure of our inference algorithm. Since we set theinitial configuration of our algorithm with homogeneous masses and a constant deterrence function, a denominatorin Eq. 2,3 is identical for all nodes so that the result of the first update stage is approximately same with that ofthe strength approximation. The Fig. 2 shows that the deterrence function of the first stage is nearly equivalent tothe result of strength approximation and approaches to the true function with repeated iteration. Thus we concludethat our inference algorithm provides an effective modification of the existing methods, and it describes a preciseembedded structure of a mobility network.

From a practical standpoint, our results for the international trade network hold important implications. First, ourinference algorithm enables to reveal the purely intrinsic capability of each country in the global economy, excludingthe external effects from surrounding countries. For instance, if a country shows a relatively low outward strengthwith respect to its outward mass, the country may export less than its production capacity because of the insufficientpurchasing power of nearby countries. Besides, our methodology allows the better inference of a deterrence functionthan the previous method as shown by the smaller error bars in Fig. 4. Further, it shows a superior performance inreconstructing the original trade flows than the prior method as verified in Fig. 6. We conjecture that those resultscan be utilized to various practical applications of trade network, such as the assessment the magnitude of potential

8

(a)

(b)

FIG. 5. Comparison of strength and mass inference for (a) outward and (b) inward information, in log-log scales.

9

a b

FIG. 6. (a) Density plot for comparison of trade flows fij from the real network and results f̄ij from our self-consistent inferenceformalism. (b) Density plot for comparison of fij and the results f̄S

ij for the simple strength-based gravity model.

trade flow between a certain country pair.

ACKNOWLEDGEMENT

DL, WC, and BJK acknowledge the support by the National Research Foundation of Korea (NRF) grant fundedby the Korean government (MSIT), Grant No. 2019R1A2C2089463.

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