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Assessing the role of migration as trade-facilitator using the statistical
mechanics of cooperative systems
Appendix A: Experimental verification of model assumptions
Our theory, developed within a classical statistical mechanical perspective, is set at a
microscopic level and it accounts for an ensemble of native `decision makers', whose behavior
(i.e., the propensity to undertake an international trade) can be affected by the interaction with
migrants. However, the theoretical outcomes of such a model are compared with available data
on the number of exporting firms. We can switch from the resolution of decision makers, where
the whole theory lies, to that of the firms, where the data analysis is performed if and only if
there exists a linear proportionality between the total population and the total amount of firms
(for otherwise amplification distortions and biases would affect the final results). This is the
case in Spain for the considered time window (1998-2012), as shown in Fig. 1. Thus, as far as
scalings are concerned, we can compare the theoretical predictions for the average behavior of
decision makers (stemming from the statistical-mechanics model) to describe the expected
attitude of firms (that we infer from empirical data).
Figure 1: The number of firms grows linearly with the overall population. Each data
point (blue bullet) represents the number of firms versus the population of a given Spanish
province (out of 50 ) for a given year (in the interval 1998 2012 ). The linear proportionality of
these quantities is highlighted by binned data (green squares), whose best fit is given by a linear
law (red solid line) with slope  1.02  0.03 and goodness R 2  0.99 .
Now, a further passage is required in order to relate the latter to the total value of trade
Y . In fact, the total value of trade Y is usually defined in terms of two contributions: the
number of firms that perform international trading (i.e. extensive margin Yext ) and the average
value (or volume) of the transactions of the trading firms (i.e. intensive margin Yint ), namely
Y = Yext  Yint , or, in a logarithmic scale, log Y = log Yext  log Yint . Chaney has shown that a
reduction in fixed trade costs has a positive impact on Yext caney; Peri and Requena have shown
that migrants have a positive effect on the extensive margin of exports in Spain, hence deriving
that migrants facilitate trade mainly by reducing the fixed costs of exporting francisco1. On the
other hand, the intensive margin of exports seems to be poorly affected by migration stocks as
shown in Fig. 2. Thus, even though our theory is actually meant to capture the evolution of Yext ,
it can be used to fit directly the evolution of the total amount of trade relationships Y as a
function of the migrant density.
Figure 2: The intensive margin of trades does not depend (in the average) on the
migrant density. Each data point (blue bullet) represents the intensive margin of trades Yint
estimated as the ratio between the overall amount of exports Y and the number of exporting
firms Yext for a given Spanish province (out of 50 ) for a given year (in the interval 1998 2012 ).
Green squares represent binned data along with the related standard deviation and the solid
line corresponds to the constant value 0.02 which best fits the binned data. The independence
of  suggests that Yint is uncorrelated with the migrant density  .
Appendix B: Insights in the statistical mechanics analysis
This appendix is meant to deepen the calculations that bring from the Hamiltonian
function
N
N N
1 1
1 1 2 
 ( , z; J ,  ) = 
(1)
J ij i j  N 1 
i  i z ,
N1 ( i , j )
i =1  =1
N
N
N
J 1
1 1 2 



(2)
 i j N 1 
i  i z ,
N1 (i , j )
i =1  =1
describing our model to the self-consistency equation
(3)
m = tanh [m(J   2 2)].
for its order parameter, which is used to infer the average behavior of the system from real
data in the section ``The model'' in the main text (see Eqs. 5-6 and Eq. 12, respectively).
We consider a system of N = N1  N2 agents, where N1 = (1   ) N denotes the
number of native agents and N 2 = N denotes the number of immigrants. Each native agent is
associated to a variable  i , with i = 1,..., N1 , representing her attitude towards trades, and

each foreign-born agent is associated to a variables z  , with  = 1,..., N2 , representing the
social capital carried by the foreign agent  . The overall configuration is distributed according
to P ( , z; J ,  ) , defined in Eq. 7 of the main text and reported here for completeness
P ( , z; J ,  ) =
exp   ( , z; J ,  )
.
Z ( , J ,  )
(4)
We now introduce the set of (migrant dependent) order parameters {m ( )} =21 as
N
N
1 1 
i  i ,
C i =1
where C normalizes with respect to the expected number of non null entries, namely
C = N1( i = 1) = N1N  =  (1   ) N 1 ,
m ( ) =

where ( i = 1) was defined as (see Eq. 4 in the main text)
(5)
(6)

( i = 1) = 1  ( i = 0) =
(7)
.
N
According to the definition (5), m ( ) is the attitude towards in international trading
for Spanish people that share the knowledge with the foreign-born agent labelled as  .
Due to the gauge-like symmetry of the model, m ( ) = m for each  = 1,, N 2 ,
where  means the average according to P . Further, as in the dilution regime of empirical
interest each decision maker  i is linked with (at least) one foreign-born agent  , we have
1
 i = m ( ) = m , i.e. m is also the averaged predisposition of the
N1 i
whole host community in international trading. This is because
that m( ) =
N
1 1
 i =  i ,
N1 i =1
m( ) =
N
1
1
N1
 i i =
 i i

C ( N ) i =1
C(N )
m ( ) =
N1

( i = 1)  i =  i .
C(N )
In terms of these new order parameters we can write
=
 2 2 (1 ) 2
2
Z (  , J ,  ) = e
(8)
N2
 


m2 ( )
=1
(9)
,

that can be evaluated straightforwardly with a standard saddle-point argument. In fact, we can
write
Z (  , J ,  ) = dm dmˆ  e

N2


e
=1
i
e
N2


mˆ  m
=1
i
e





mˆ m ( )
=1
 2 2 (1 ) 2
= dm dmˆ  e

2
N2



m2
=1

 N2

 i
mˆ   

  =1
N1 log2  N1 logcosh
  (1 ) N 1





N2
i
2

N2
2
m
 2 2 (1 ) 2



mˆ m











e
e
,
which, in the large N limit, can be rewritten as
=1
(10)
Z (  , J ,  ) = dm dmˆ  e

 e
N1 ({ m },{mˆ  })
(11)

ˆ })
N1
 ({m },{m
sup

{m },{mˆ  }
,
where
N2

  i mˆ   

 =1
 ({m },{mˆ  }) = log 2   logcosh 
1
  (1   ) N


 2 2 (1   ) 2
N2







N
i 2

m 

mˆ  m .
2 N1
N1  =1
 =1
Taking the sup of  we get the following set of self-consistent relations
 m  ({m },{mˆ  }) = 0 
2

i
 2 2 (1   ) 2
mˆ  = 
m   imˆ  =  2 2 (1   ) 2 m
N1
N1
 mˆ  ({m },{mˆ  }) = 0 

N2




 i mˆ    

i
i


 =1
m =
   tanh 
,
1
1 


N1
 (1   ) N
 (1   ) N






that, once solved together, returns the value of the order parameter as a solution of
  
m =   
N 
1

 tanh (  (1   )N
2
 1
N2
m   )


=1
.
(12)
.
(13)


Handling the average over  , indicated with   , we get
N2



m =   tanh   2 (1   )N  1  m  m   


 




Now we can look for the solution m = m for each  : deleting the vanishing term N  1m (that
goes to zero in the thermodynamic limit) we find that m obeys
m = tanh [  2 (1   )m] ,
(14)

where we defined the random variable  = N  1  2   N  1  =21  .
N
N
Evaluating the first and second momenta of  is straightforward and they read,
respectively, as
[ ] = N  1 N 2 [  ] =  ,
N 
Var[ ] = N 2( 1) N 2Var[  ] =  ( N 2( 1) NN  )  0.
This means that, in the limit of infinite size, the variable  is deterministic and thus we have
(15)
m = tanh [ 2 2 (1   )m].
This formula relates the expected amount of exporting firms m (and, similarly, the expected
value of international exports) with the fraction  of foreign-born people in the province
considered.
The full Hamiltonian (1) also contains an intra-party interaction term encoded by J ,
which was not considered in this treatment. Accounting also for this contribution simply implies
an additional term Jm , being J the average of the entries J ij of J over all possible couples,
in the argument of the hyperbolic tangent of Eq. 15, namely
m = tanh (Jm   2 2m),
where we used  =  (1   ) .
(16)
Appendix C: A corroboration of the model from a graph-theory
perspective
As shown in the main text, our bipartite model can be mapped into a monopartite
network for natives only. In the real world, social networks (i.e., the structures of relationships
between social entities, such as persons, groups, organizations or countries) typically exhibit
small-world features, namely a small diameter (i.e., the distance between any pair of nodes is
relatively small) and a large clustering (i.e., the neighbors of a node are likely to be neighbors
themselves) Newman-2010,vespignani. Reliable models for social networks are therefore
expected to recover such properties. Thus, if we can show that the emerging monopartite
network is small-world, we have an additional (beyond the successful fits with empirical data on
trade relationships) and independent corroboration of the validity of our model. This is the aim
of this appendix.
Before proceeding it is worth introducing one of the most popular random network
model, that is, the Erdös and Rényi network (ER network): given a fixed vertex set, edges are
included randomly, each with probability p independent of every other edge. This type of
random network has a small diameter (it grows logarithmically with the system size) in such a
way that the path between a pair of nodes typically involves only a few edges (this notion has
been popularized by terms like the `six degrees of separation' Newman-2010). This property is
fundamental in order to spread information over the network fast (i.e., in few passages).
However, ER networks do not display a large clustering: the probability for two nodes to be
linked is always p , independent of whether they share a common neighbor. Therefore, the ER
network is not suitable to mimic social structures, yet it constitutes a good model to test the
clustering of a given graph: fixing the size N and the average number d of neighbors per
node, if in the graph under study the ratio between the number of triangles and the number of
pairs sharing a common neighbor is larger than d/N (i.e., the value expected for an ER graph),
then it can be looked at as highly clustered.
In the context of information spreading, a large clustering constitutes a positive
element as it implies a large degree of redundancy (to contrast decay effects - the novelty of a
message usually tends to fade with time and hence the attention people pay to it), a social
reinforcement (individual's selection of message items can be naturally expedited by the
increasing frequencies of the same choices of other people), and a relative ease in reaching all
agents in the network vespignani. Thus, as for our model, the high clustering is a crucial
requirement because it favors the spreading of the social capital, namely the information
supplied by an immigrant about her country of origin to the decision makers about trading
abroad in the host country.
As recalled above, in our model the community under investigation is split into natives
and immigrants and their mutual interaction (through which the exchange of information
occurs) is described in terms of a bipartite graph (see Fig. 1 , left panel). The related statistical
mechanics analysis shows that if the local agents i and j both interact with some foreign-born
individual  , i.e. i j  0 , then the agents i and j can be thought of as directly interacting
~
via an effective coupling J ij  i j (in the main text see Fig. 1 , right panel). Otherwise
stated, what emerges from our model is that sharing a large number of acquaintances among
immigrants is indicative of closeness: if two agents display a large number of common
acquaintances, they are probably related somehow (i.e. colleagues, friends) and this pair of
agents will be connected. Therefore, through this mapping we infer a picture of the social
network constituted by decision makers.Ê We now consider such emergent mono-partite,
weighted network, referred to as  , and review its topological properties (former
mathematical investigations can be found in Agliari-EPL2011,Barra-JStat2011,BarraPhysA2012), focusing on those known to characterize real social networks.
A global characterization of the graph  can be attained in terms of the average link
probability plink : considering a generic couple of nodes, say i and j , keeping a mean-field
perspective, we can write
N2

plink = 1   1  ( i = 1)( j = 1)
 =1

(17)
N

2 
= 1  1  2  ,
(18)
 N 
where in Eq. (17) the term in the square brackets represents the probability that the
N
~
contribution i j in the sum J ij =  =21i j N12( 1) (see Eq. 10 in the main text) is equal to
zero and the product over  returns the probability that all entries  = 1,..., N2 are null such
that, finally, the complementary of this quantity provides the probability that at least one entry
~
is non-null, that is, that J ij > 0 ; in Eq. (18) we used the homogeneity of pattern entries in eq.
(7) and the definition of  as   N2 /N (see also Eq. 1 in the main text). The average number of
links stemming from a node (also known as the degree of the node) in  therefore reads as
d = plink N1 .
Now, as  and  are tuned, the emerging graph can range from fully connected to
completely disconnected Agliari-EPL2011,Barra-JStat2011 and we can distinguish the following
topological regimes (whose examples are reported in Fig. 3):
  < 1/2 , plink  1, d  N  Fully connected (weighted) graph.
  = 1/2 , plink  1  e   , d =  ( N )  Linearly extensive degree.
2
1/2 <  < 1,

plink   2N 1 2 ,
d =  ( N 2(1 ) )

Extreme dilution regime:
1
lim N  d = lim N  d /N = 0 .
plink   2/N ,
 Sparse (weighted) graph;  2 = 1
  = 1,
d =  (N 0 )
corresponds to the percolation threshold.
Summarizing, large values of  determine a disconnected graph with vanishing average
degree. Therefore,  coarsely controls the connectivity regime of the network, while  and 
allow a finer tuning.
Figure 3: Examples of network corresponding to different regimes of connectivity. (a)
Fully connected network; (b) Linearly extensive degree; (c) Extremely diluted network; (d)
Sparse network.
For the Spanish population under consideration we can get an estimate of the value of
these parameters using empirical data.
As for  , this can be obtained directly from its definition  = N2 /N , by comparing the
foreign-born population with the total population and it ranges in  (10 3 )  (10 1 ) .
As for  , we can derive it through an indirect measure: we expect that the number of
links between natives and immigrants is lower bounded by the number of mixed marriages
M mixed . In fact, a mixed marriage yields, in general, several `mixed acquaintances' between the
family members and the friends of the two parties.
~
~
We can write the probability pmixed of mixed marriage as pmixed  N  , with    ,
therefore, we can also write
M mixed  N1  N 2  pmixed  N1  N 2 

~
N
,
(19)
from which
~
M mixed
(20)
 N 1 .
N2
Mixed marriages in Spain have been thoroughly investigated in Barra-ScRep, Agliari-NJP and
~
~
from those data we can fit the ratio M mixed /N 2  N 1 inferring an estimate for  , hence the
second dataset we use (social data) is built from statistics over residential variation in Spanish
municipalities, the so called estadistica de variaciones residenciales (EVR) Barra-ScRep. As
shown in Fig. 4, the number of normalized mixed marriages is roughly constant with respect to
~
N , that is   1 . As a consequence,   1 . On the other side, a value of  strictly smaller that
1 would imply that the number of connections between the two parties grows indefinitely with
N (or, analogously, with N1 or N 2 ), and this is certainly not realistic Barra-JSP2008,bianconi.
Thus, the experimental argument for the lower bound coupled with the theoretical argument
for the upper bound imply  = 1 (as intuitive).
Figure 4: The number of mixed marriages divided by the foreign population does not
depend (in the average) on the overall population. Data on mixed marriages for each province
along the years 1998-2012 are drawn from the local offices of Vital Records and Statistics
(Registro Civil) and divided by the related size N 2 of the immigrant community. Raw data (blue
bullets) are properly binned (green squares) to highlight the effective behavior with respect to
the overall size N of the related province. The red line shows the lack of dependence on N ,
for M mixed /N 2 , in the large N limit.
Finally, we need to estimate  . According to Eq. 7, and having fixed  = 1 , 
represents the average number of local acquaintances displayed by an immigrant. In our
analysis we bound  in between 5 (we expect that any immigrant has at least five links with
the local community) and 25 : there are several sociological studies trying to estimate the
average number of acquaintances (familiars and/or friends) of a member of societies. In
particular, in Estudio,HumanOrganization this analysis is performed in Spain finding that this
number is  =  (10) , similarly to other European countries.
According to these estimates for  ,  and  , we get that  is a sparse graph BarraPhysA2012,Agliari-PRE2011, namely a graph with an average degree that does not scale with
the system size.
Also,  turns out to be highly clustered since the ratio between the average clustering
coefficient c(,  ) measured in a numerical realization of  and the average clustering
coefficient cER of an ER network displaying the same average degree is larger than 1 From
~
another perspective, one can notice that the definition for J ij , namely the Hebbian kernel (see
Eq. 10 in the main text), implicitly endows couplings with ``transitivity'': if i and j are
connected as they share acquaintances among immigrants, and the same holds for i and z ,
then j and z are also likely to share any acquaintance. Otherwise stated, interactions based
on sharing (i.e., matching non-null entries) intrinsically generate a clustered society. (see also
Agliari-EPL2011,Barra-JStat2011,Barra-PhysA2012). Here we further deepen how c(,  ) varies
as a function of  and  , the latter being tuned within the ranges empirically detected as
described above. As shown in Fig. 5 (left panel), in general c(,  ) depends non monotonically
on  and  . The reason is that for relatively small values of  a small fraction of nodes is
isolated, while the connected component of  is very tightly connected. When  is increased,
isolated nodes tend to get (loosely) connected in such a way that the overall clustering
coefficient is reduced. By further increasing  , more and more links are inserted and the
clustering coefficient consistently grows Agliari-PRE2011. However, in the region where Y
depends more sensitively on  (namely  < 0.1, see Fig. 3 in the main text) a growth in  or
in  typically corresponds to a growth in the clustering coefficient and therefore in an
improvement in the network ability to spread out the information.
As for the diameter, we find that it grows slowly with the system size as expected for
small-world systems. In particular, by numerical realizations of  , we get that the diameter
d (,  ) exhibits a slight dependence on  and, fixing  = 10 , as the size N is varied from
O(10 2 ) to O(10 4 ) , the diameter d (,  ) only changes logarithmically from 3 to 10 (see Fig. 5,
right panel).
Therefore in the parameter region evidenced empirically our model successfully mimics
the expected social structure.
Moreover, we notice that in order to improve social reinforcement and get a more
reliable spreading of information throughout the network, policies should be designed to
increase cross-links. Interestingly, this strategy agrees with the one obtained from the statistical
mechanics perspective, because a large value of  implies a reduction in the threshold c (see
Eq. 16 in the main text). These features are two sides of the same coin: a large clustering
coefficient, favoring the spreading of information throughout the network, optimizes the
fruition of the information provided by immigrants, in such a way that, in order to boost trade
relationships, a smaller (with respect to a random graph) fraction of migrants is necessary.
Figure 5: The clustering coefficient and the diameter of the emerging graph for natives
only are evaluated in the region of the parameter space which is empirically sound and smallworld features are evidenced. Left panel: Clustering coefficient c(,  ) as a function of  and
 for a system of size N = 2000 and  = 1 simulated numerically. A high degree of clustering is
reached when  and  are relatively large. Right panel: Diameter d (,  ) as a function of 
and  for a system of size N = 2000 and  = 1 simulated numerically. A small diameter is
obtained when  is large; small values of  also give rise to a small diameter but this is related
to the fact that the graph is fragmented into several components. In the overpercolated region
d depends smoothly on the system parameters and we checked that there it grows
logarithmically with the system size. Therefore, values of  and  consistent with those found
empirically correspond to a small-world structure and by further increasing  and/or  these
features are even more emphasized.
Finally, recalling that  is a weighted graph, one can check for consistency with real
world networks also beyond the bare topology, namely as for the weight distribution is
concerned. Here we just mention that according to the strength of weak ties theory by
Granovetter Granovetter-AJS1973, Granovetter-ST1983, the degree of overlap of two
individuals' neighbourhoods varies directly with the strength of their tie to one another (i.e., if
the two individuals are acquaintances rather than close friends, there is little overlap) and weak
ties also turn out to be crucial in order to maintain the network connected. Consistently, in the
graph  weak ties connect individuals sharing a small number (possibly only one) of
connections in the immigrant community and, by cutting (a relatively small number of) weak
links, the network gets fragmented into several components Barra-PhysA2012,Agliari-
PRE2011,Callaway-PRL2000,Barabasi-RevModPhys2002,marti,vespignani.