To conclude, aggregation of the two versions of the database has been made, the problem of inconsistency in observations has been resolved, loops have been eliminated and as a result, the annual data on international migration flows from 1970 to 2013 for 215 countries has been obtained.

2.2 Centrality indices interpretation

International migration patterns are usually analyzed by simple measures as number of migrant inflows and outflows, net and gross migration flows. These measures are basic and can be useful for a certain country in application for its migration policy. However, global migration forms the network of countries and all of them are interconnected through migration flows. Therefore, in our analysis international migration is modeled as a graph, where nodes are countries and edges are migration flows.

We study the properties of international migration flows from the network prospective, evaluating the centrality indices. The aim of this methodology is to provide a ranking of the vertices based on their importance for the system.

First, we introduce the application of classical centrality indices to international migration. Second, we apply new centrality indices, which have certain distinctive features in comparison with classical centrality indices.

2.2.1 Classical centrality indices

In our work the following centrality measures are evaluated: degree and weighted degree centrality, closeness, eigenvector and PageRank.

The degree centrality is the number of nodes each node is connected with [20]. For directed graph the degree centrality has three forms: the degree, in-degree and out-degree centrality. The in-degree centrality represents the number of in-coming ties each node has, and out-degree is the number of out-going ties for each node.

In terms of migration, edge in unweighted graph characterizes the presence of migration flow between any two countries. The in-degree centrality for country A is the number of countries, which are connected with country A through migration in-flows to country A. In other words, it is the number of countries, which migrants came from to country A. For out-degree the interpretation is as follows: the number of countries which are connected with country A through migrant out-flows from A or the number of countries which are the destinations of migrants from A. The degree centrality of country A can show how many different countries are connected with A through migration flows.

The following centrality indices are estimated for the weighted network: weighted in-degree, weighted out-degree, weighted degree difference (=weighted in-degree - weighted out-degree) and weighted degree [20]. The weighted in-degree centrality represents the number of in-coming ties for each node with weights on them or the immigrant flow to the country. Weighted out-degree is the number of out-coming links for each node and accordingly relates to the number of emigrants. The weighted degree difference is the difference between migrant in-flow and out-flow which is the net migration flow. The weighted degree is the sum of weighted in-degree and weighted out-degree centralities for each country or the total number of emigrants and immigrants (gross migration). These centrality indices can give us the basic information about the international migration network: the level of migrant in-flows and out-flows, net and gross migration flows.

The closeness centrality [9] shows how close node is located to the other nodes in the network. In addition, this measure has the following characteristics. Firstly, it accounts only for short paths between nodes. Secondly, these centralities have very close values and are sensitive to the changes in network structure: minor changes in the structure of network can lead to significant differences in raking by this centrality. In our work this centrality is estimated for the undirected graph with maximization of the weights on paths and is related to the level of closeness of particular country to intense migration flows. Note that it does not imply that the country itself should have huge migration in-flows or out-flows. This measure can provide the information about potential migration flow to particular country by estimation the distance between the country and countries with huge migration flows in the network. Countries with low closeness centrality are not necessarily involved in the process of international migration since they usually have low migration in-flows and out-flows.

Eigenvector [12] is the generalized degree centrality, which accounts for degrees of node neighbors. Both Eigenvector [12] and PageRank [13] centralities measure the influence of a node in the network. The idea of these indices is that the particular node has a high importance if nodes linked with it have a high importance In international migration network these indices highlight the countries - “centers of international immigration”, and the countries, which are directly linked with them through migration flows.

2.2.2 Short-Range Interaction and Long-Range Interaction Centrality indices

Short-range interaction centrality index (SRIC) is based on the power indices proposed in [5] and applied for defining systemically important financial institutions or countries in [6]. The key difference of this index from classic centrality indices is that indirect influence between countries in the network are taken into account and another node attribute is being presented - the population of destination country.

The indirect influence of country A to country B through migration flows is important to be considered in the network analysis for two following reasons. First, migration between any two countries may occur not directly, i.e. there can be a migration route. In this case the understanding of country with highest indirect influence, i.e. the initial country that is generating the migration flow is meaningful to highlight the most powerful countries in the global migration network. Second, as all countries in international migration network are interconnected, the flow between any two countries can lead to emergence of new flows between any other countries. In this case flows of migrants do not necessarily consist of the same people, as we do not know migrants' characteristics (nationality, gender and other). Both cases are possible in the analysis of indirect influence of countries in the network. However, classic centrality indices do not consider the indirect interactions. Therefore, it is relevant to estimate the Short-range and Long-Range Interactions Centrality indices.

We evaluate the influence of migration flows to the population of destination country through imposing the quota by population for destination country. We suggest that 0,1% of population of destination country is the critical level of migrant inflow. If migration flow from country A to country B does not reach 0,1% of population of country B, then country A does not have influence on country B through migration flows.

The critical group of countries is interpreted as a group whose total number of migrants is crucial in terms of quota for the population of destination country, i.e. the group is critical if the total amount of its members' immigrants is greater than or equal to a predefined quota. Country is pivotal in the critical group, if without this country group is no longer critical. The intensity of connections is estimated by the following formula

, (2.1)

where is a critical group of countries with respect to a country A (country of origin), in which a country B (destination country) is pivotal, is the total number of migrants came from country A directly to B, is the total number of migrants came from country A to B indirectly - via any other country. Below a simple example is presented of the different indirect paths from country A to country B₃ for the short-range interaction index.

Figure 2.1.

As we can see from the graph, there are three different ways to reach B₃ from A: 1) A-B₁-B₃, 2) A-B₂-B₃ and 3) A-B₁-B₄-B₂-B₃. SRIC accounts only for the first order connections as in the cases 1) and 2). However, migrants from A can move to B₃ using longer route, as in 3) example and in this case we need to re-evaluate the estimation of index to consider s-long-range routs (in this case 3-long range route).

Long-range interaction centrality index (LRIC) is estimated by the following methodology.

Firstly, the matrix of bilateral migration flows is constructed , where is the migration flow from country i to country j. Then we construct a matrix with respect to the matrix A and predefined quota as

(2.2)

where is a critical group of direct neighbors for the element i,, and is critical group for the element i,. A group of neighbors of node i is critical if .

Obviously, the construction of matrix is highly related to [6] because it requires to consider separately each element of the system as a country of origin while other participants of the system are assumed as destination countries. The only difference here is that in our approach only groups of direct neighbors are considered.

The interpretation of matrix C is rather simple. If , then the destination country j has a maximum influence to the country of origin i. On the contrary, if then the country of origin j does not directly influence the country of destination i. Finally, the value indicates the level of impact of the origin country j on the destination country i.

Thus, we evaluate the direct influence of the first level of each element in the system. To define the indirect influence between two elements let us give a definition of the с-path.

Denote by с a binary relation which is constructed as .

A pair (i, j) such that iсj is called a с-step. A path from i to j is an ordered sequence of steps starting at i and ending at j, such that the second element in each step coincides with the first element of the next step. If all steps in a path belong to the same relation с, we call it с-path, i.e., a с-path is an ordered sequence of elements i, j₁, …, j_k, j, such that iсj₁, j₁сj₂, …, j_k-1сj_k, j_kсj. The number of steps in a path is called the path's length.

To define the indirect influence between any two elements consider all с-paths between them of length less than some parameter s. We consider only paths with no cycles, i.e. there are no elements that occur in the с-path at least twice.

Denote by a set of unique с-paths from i to j, where m is the total number of paths and denote by , where , a length of the k-th path. Then we can define the indirect influence between elements i and j via the k-th с-path as

, (2.3)

or

, (2.4)

where , is an l-th element which occurs on k-th с-path from i to j.

The interpretation of formulae (2.3) and (2.4) is the following. According to the formula (2.3) the total influence of the element j to the element i via the k-th с-path is calculated as the aggregate value of direct influences between elements which are on the k-th с-path between i and j while the formula (2.4) defines the total influence as the minimum direct influence between any elements from the k-th с-path.

It is necessary to mention that in some cases there is no need to consider all possible paths between elements i and j, i.e. we can assume that starting from some path's length s indirect interactions does not influence the initial member. Thus, we design the parameter s that defines how many layers (path's length) are taken into account.

Since there can be many paths between two elements of the system, there is a problem of aggregating the influence of different paths. To estimate this aggregated indirect influence several methods are proposed.

The aggregated results will form a new matrix .

1. The indirect influence: sum of paths influence

. (2.5)

2. The indirect influence: maximal path influence

. (2.6)

The interpretation of formulae (2.5)-(2.6) in terms of migration flows is the following. The sum of paths influences gives the most pessimistic evaluation of the indirect influence where we take into account all possible channels of risk from a particular destination country to the country of origin.

Thus, we can define the indirect influence between elements i and j via all possible paths between them. The paths influences can be evaluated by formulae (2.3)-(2.4) and aggregated into a single value by formulae (2.5)-(2.6). Thus, four combinations are possible for matrix construction (see Table 2.2). In our opinion, all possible combinations of the formulae have a sense except the combination of formulae (2.4) and (2.5).

Table 2.2. Possible combinations of methods for indirect influence

	Paths aggregation
	Sum of paths influence	Maximal path influence
Path influence	Multiplication of direct influence	SumPaths	MaxPath
	Minimal direct influence	-	MinMax

The aggregation of matrix into a single vector that shows the total influence of each element of the system can be done with respect to the weights (importance) of each element as it is done in [6].

To sum up, the classic centrality indices and indices of short and long-range interactions are applied to characterize the countries in migration network. The distinctive feature of the latter is the consideration of the population of destination country and indirect migration routes between countries.

2.3 Results

The centrality indices are evaluated for each year of the period 1970-2013. The analysis for each decade is presented in the following form. First, the overall picture of the international migration in the corresponding decade is observed by overview of the major migration corridors. Second, the results of evaluation of classic centralities and SRIC, LRIC indices are presented. Finally, the comparison of results is made by performing the correlation analysis.

1970-1979

The major migration corridors for 1970s occurred between Turkey and Germany (in both directions), Yugoslavia and Germany (in both directions), within the European countries, from Mexico to the USA and from the UK to Australia. The migration ties between developed European countries and developing countries during this period is explained by the labor migration program [35]. This program influenced the migrant inflow from south European countries (Italy, Spain and Greece) and developing countries outside European region (Turkey). The situation changed after the oil crisis in 1973. The guest labor migration program was over and it caused the emigration of people, which already were unemployed. Additionally, in this decade the migration flow from Mexico to the USA begins to exceed 50 000 of migrants since 1972. On the contrary, migration from the UK to Australia drops and after 1974 is no longer presented in the list of corridors over 50 thousand of migrants. Now we present the results of evaluation of centrality indices. As was mentioned above, major migration corridors did not change considerably, hence centrality indices did not differ a lot during these years. To represent the international migration flows in the 1970s from the perspective of centrality indices the 1972 results are chosen. To begin with, migration flows over 50 thousands for 1972 are presented in Table 2.3. This list complies the major migration corridors occurred in the 1970-1979. Consequently, let us provide the ranking of countries by the centrality indices (Table 2.4).

Table 2.3. Migration flows over 50 000 in 1972

Origin	Destination	Migration flow
Turkey	Germany	161 430
Germany	Italy	122 888
Germany	Turkey	111 401
Germany	Yugoslavia (former)	102 588
Italy	Germany	88 062
Yugoslavia (former)	Germany	72 835
Mexico	USA	71 586
UK	Australia	63 800

Table 2.4. Rankings by centrality indices for 1972

Country	WInDeg	WOutDeg	WDeg	WDegDiff	Clos	PageRank	EigenVec	SRIC	LRIC (SUM)	LRIC (MAX)	LRIC (MAXMIN)
Germany	1	1	1	2	2	1	1	1	5	4	4
USA	2	6	2	1	1	2	6	2	4	6	8
Italy	3	3	3	212	5	4	2	4	3	3	3
Yugoslavia (former)	4	5	5	211	6	6	3	6	1	1	1
UK	5	4	6	213	13	3	9	7	9	11	12
Canada	6	18	7	3	7	5	12	11	14	17	15
Turkey	7	2	4	215	3	7	4	3	2	2	2
Australia	8	12	10	4	25	8	16	5	12	12	11
Greece	9	7	8	205	10	9	5	16	6	5	5
Spain	10	9	9	198	11	11	7	12	7	7	6
Netherlands	11	15	11	7	9	12	10	15	13	13	13
Belgium	12	16	12	9	16	10	13	19	22	21	16
Sweden	13	13	13	199	4	13	19	8	17	20	27
Austria	14	11	14	202	14	14	8	18	8	8	7
France	15	14	16	204	17	16	11	10	10	9	9
South Africa	16	27	20	5	31	15	18	13	20	19	18
Finland	17	24	18	8	24	18	23	9	18	18	25
New Zealand	18	31	23	6	40	17	32	14	25	24	23
Portugal	20	10	17	210	15	20	15	17	11	10	10
Norway	23	47	36	10	49	23	33	41	41	41	41
Mexico	42	8	15	214	8	42	45	41	41	41	41

Germany was involved in the largest migration flows as migrant destination and origin country. Therefore, it has highest weighted in-degree centrality (migrant inflow), weighted out-degree (migrant outflow), weighted degree (gross migration flow), correspondingly. Weighted in-degree centrality also highlights the immigration countries: Italy, Yugoslavia (former) and the UK.

Weighted out-degree centrality results correspond to the countries-suppliers of the labor force - Turkey and Italy. The UK is in top of countries by this centrality because of the flow to Australia.

The highest weighted degree centrality or the gross migration flow have most involved into the process of international migration countries (Germany, the USA, Italy, Turkey and Yugoslavia (former)). The USA are constantly ranked the first by weighted degree difference (net migration flow). This fact is explained not only by the attractiveness of this country for migrants, but that the USA do not provide the emigration statistics, hence net migration flow is underestimated.

Closeness centrality ranks the countries based on the presence of connections with main migrants' origin or destination countries. The new country introduced by this centrality is Sweden, because there was emigration from Sweden to both the USA and Germany.

PageRank and Eigenvector centralities account for attractive migrants' destination countries (Germany, the USA, Yugoslavia (former) and Italy) and, in addition, countries connected with them (the UK, Canada and Turkey).

Overall, ranking by classical centrality indices shows the countries directly involved in the process of international migration: top countries of migrants' destination, origin and their direct neighbors in the network.

Consideration of the indirect interactions can help to outline a new list of countries with high influence in international migration network.

SRIC ranking of countries is highly related with ranking by weighted in-degree centrality. However, Turkey is presented among top three countries, and Finland appears in top ten. The explanation of these results can be the following. Turkey has direct connections with Germany, through highest migration inflow and outflow. Finland also has migrant inflow and outflow to Germany, nonetheless they are not the massive (2 862 and 3 663, correspondingly). They influence Finland, because population of Finland was not very large (4 639 657) in comparison with other countries.

Each LRIC index highlights Yugoslavia (former) and Turkey, as these countries are interconnected with the countries - centers of migrant attraction (Germany, the USA, Italy), which have lower position in ranking. Interestingly, Greece is outlined in top six countries. Greece had both immigrants from Germany (from Germany to Greece 48 538) and sent migrants to Germany (51 509), the USA (11 021) and Canada (4 016). Additionally, population of Greece was 8 888 628 in 1972. Spain and Austria also had higher ranking by LRIC indices. As in the previous cases consideration of indirect interactions of these countries in the network and their population has made them rise in ranking. Spain was a labor supply country for Germany, therefore they were connected by both inflows and outflows of migrants in 1972. Austria and Germany had established migration connections because of geographical and cultural proximity.

SRIC and LRIC indices define different from classical centralities rankings of countries. These indices outline not only top migrant origin and destination countries, but also the countries connected with them (Greece, Spain, Austria) and countries, where immigrants have considerable share of the population (Finland).

For comparison of rankings of countries by different centralities, correlation analysis is applied. As the position of country in the ranking is the rank variable Goodman, Kruskal -coefficient [22] was estimated for each year of the period. The results did not vary considerably for each year. Therefore, the estimation results are provided for 1972 (Table 2.5) as an example.

Table 2.5. Goodman, Kruskal -coefficient for 1972

	SRIC	LRIC (SUM)	LRIC (MAX)	LRIC (MAXMIN)
WInDeg	0.91	0.918	0.918	0.908
WOutDeg	0.874	0.881	0.88	0.877
WDeg	0.889	0.89	0.89	0.885
	SRIC	LRIC (SUM)	LRIC (MAX)	LRIC (MAXMIN)
WDegDiff	-0.392	-0.401	-0.4	-0.401
Clos	0.885	0.888	0.887	0.882
PageRank	0.9	0.907	0.907	0.898
Eigenvec	0.897	0.92	0.921	0.91
SRIC	1	0.966	0.963	0.97
LRIC (SUM)		1	0.995	0.984
LRIC (MAX)			1	0.983
LRIC (MAXMIN)				1

The ranking by SRIC and LRIC is highly related to eigenvector, PageRank and weighted degree centralities, as was observed after the estimation of Goodman-Kruskal correlation coefficient [22]. Additionally, SRIC and all LRIC indices are highly correlated between each other and weakly with weighted degree difference. However, as it is mentioned in the description of the results above classical centrality indices do not consider countries connected with top migrant destinations and share of the migrants in the population of the country.

1980-1989

The international migration flows during this decade can be divided into the following groups: 1) from Central America to the USA, 2) from Southeast Asia to the USA, 3) intra-European migration, 4) from Turkey and Yugoslavia (former) to Germany.

Migration flows to the USA from the Central American countries were characterized by the rise of the inflow from Mexico and the development of the new flows from other countries of this region (El Salvador). Also, the inflow from the Southeast Asia countries that already occurred in the previous decade became more intense. It represents the immigration of qualified labor force, which receives higher education in their country of origin (the Philippines, Vietnam) and migrate to the USA to provide their families with remittances.

The flows already established in the previous period - from Turkey and Yugoslavia (former) to Germany - are still presented and there is a considerable rise in Poland to Germany migration caused by the economic and political crisis in Poland.

The situation changed in the end of the 1980s: new migration flows were generated by Germany reunification and economic and political crisis in the USSR in 1989. The first caused the migration corridor between German Democratic Republic (former) and German Federal Republic. The USSR crisis in economy and politics was officially declared in 1989 and caused the wave of emigration to Germany.

In Table 2.6 below the major migration flows for 1989 are provided.

Table 2.6. Migration flows over 50 000 in 1989

Origin	Destination	Migration flow
Poland	Germany	455 075
Mexico	USA	405 172
German Democratic Republic	Germany	388 396
Germany	Poland	145 903
USSR (former)	Germany	121 378
Turkey	Germany	86 643
Yugoslavia (former)	Germany	63 438
El Salvador	USA	57 878
Philippines	USA	57 034

Considering the overview of the migration flows mentioned above it is interesting to view how these processes are described by the centrality indices. Till the end of the decade the ranking of countries by centralities has not change a lot. At the end of the decade because of the changes in international migration mentioned above the results of centrality indices have also evolved dramatically for 1989 (Table 2.7).

Table 2.7. Rankings by centrality indices for 1989

Country	WInDeg	WOutDeg	WDeg	WDegDiff	Clos	PageRank	EigenVec	SRIC	LRIC (SUM)	LRIC (MAX)	LRIC (MAXMIN)
Germany	1	2	1	2	10	1	1	2	7	6	6
USA	2	8	2	1	2	2	3	9	4	7	7
UK	3	4	4	5	16	3	7	6	12	13	16
Australia	4	6	6	4	9	4	11	7	10	16	14
Canada	5	27	7	3	1	6	8	35	29	43	35
Poland	6	1	3	214	11	5	2	3	5	4	4
Netherlands	7	23	11	6	3	7	12	29	22	27	36
Italy	8	11	9	8	18	9	6	15	8	9	8
Sweden	9	40	15	7	8	10	24	13	44	50	52
New Zealand	10	14	13	182	6	8	32	12	15	14	13
Turkey	11	7	8	211	20	12	4	5	2	2	2
Yugoslavia (former)	12	10	12	203	29	14	5	16	6	5	5
Denmark	13	24	19	10	4	11	28	18	42	46	48
Spain	14	37	22	9	17	16	20	30	25	29	44
France	16	16	16	199	28	18	9	17	9	15	20
Norway	17	34	25	111	5	17	35	10	43	40	50
Ireland	18	12	14	207	30	13	22	4	11	8	9
Austria	21	38	33	158	39	23	10	51	28	18	21
USSR (former)	24	5	10	213	19	25	15	8	3	3	3
Russian Federation	25	26	30	196	194	27	14	44	18	10	10
Philippines	44	9	17	212	21	44	44	11	21	22	18
Mexico	66	3	5	215	7	67	52	1	1	1	1

Countries with the highest inflow of migrants - Germany, the USA, the UK and Australia are the leaders in weighted in-degree, weighted degree and weighted degree difference. From the ranking by weighted out-degree centrality the following countries with largest outflow of migrants are presented: Poland, Germany, Mexico, the UK and the USSR (former). Weighted Degree centrality do not outline any new countries. The largest net migration flow had Germany, the USA, Australia and the UK.

Canada, the USA, Netherlands and Denmark have the highest ranking by closeness centrality. The USA are in the top of ranking because of the huge inflows. Other countries had outflows of migrants to the USA or to Germany, both were the international immigration centers.

Top ten countries by PageRank centrality are almost the same as top ten countries with highest migrant inflow. Germany, Poland, the USA, Turkey and Yugoslavia (former) have the highest eigenvector centrality, as they are involved in international migration by having huge migration flows and are interconnected with each other. Additionally, France is observed in top ten ranking. France has both immigrants from Germany and emigration flows to this country.

Classical centrality indices have outlined countries involved in mass migration flows and countries with direct flows to or from the international migration centers, as in the previous decade. Different observations are made after the analysis of SRIC and LRIC results.

Ireland is included in the ranking by SRIC index. It is a new country in the ranking, as it is not highlighted by classical centrality indices. Ireland is a country with the population of around 3.5 million and there was an inflow from the UK of 14 200 migrants, which exceeded 0,1% of population of Ireland (share of immigrants reached 0.4%). Mexico, Turkey, the USSR (former), Poland and Yugoslavia (former) have the highest LRIC ranking, because of their interconnections with countries of huge migration flows.

An evaluation of SRIC and LRIC indices for 1989 contributes by presenting the countries with high share of immigrants (Ireland) and countries with huge emigration to popular migrants' destinations. Results of Short and Long-Interactions Centrality indices are highly related to weighted out-degree centrality and PageRank as in the previous decade. Correlation coefficient (-coefficient) confirms these comparisons (Table 2.8).

Table 2.8. Goodman, Kruskal -coefficient for 1989

	SRIC	LRIC (SUM)	LRIC (MAX)	LRIC (MAXMIN)
WInDeg	0.697	0.731	0.719	0.707
WOutDeg	0.818	0.795	0.789	0.781
WDeg	0.82	0.799	0.792	0.783
WDegDiff	-0.581	-0.541	-0.545	-0.543
Clos	0.711	0.673	0.661	0.653
PageRank	0.699	0.737	0.723	0.713
Eigenvec	0.674	0.693	0.585	0.71
SRIC	1	0.892	0.891	0.883
LRIC (SUM)		1	0.972	0.95
LRIC (MAX)			1	0.961
LRIC (MAXMIN)				1

1990-1999

International migration in the last decade of the 20^th century was characterized by the huge migrant inflows to the USA (from Mexico and Southeast Asia) and appearance of the new international migration flows from the former Soviet Union (fSU) countries and within them due to the collapse of the Soviet Union. Already established migration flows from Yugoslavia to Germany and from Turkey to Germany were still among the mass migration corridors. Additionally, Germany was the second destination country for the migrants from the fSU countries (after these countries themselves). The next group of migration flows were intra-European migration flows, predominantly from Eastern to Western European countries.

The international migration flows were the most multidimensional and massive in 1992, year after the collapse of the USSR and stabilized till the end of the decade.

In order to explore the year of the most intensive migration in the decade by analysis of the centrality indices the international migration flows in 1992 are chosen. The amount of migration flows over 50 000 of migrants in 1992 (Table 2.9) is the highest for the whole period from 1970 to 2013.

Table 2.9. Migration flows over 50 000 in 1992

Origin	Destination	Migration flow
Russian Federation	Ukraine	309 336
Yugoslavia (former)	Germany	267 000
Mexico	USA	213 802
Ukraine	Russian Federation	199 355
Kazakhstan	Russian Federation	183 891
Poland	Germany	143 709
Uzbekistan	Russian Federation	112 442
Germany	Poland	112 062
Germany	Yugoslavia (former)	95 720
Russian Federation	Kazakhstan	87 272
Kazakhstan	Germany	86 864
Russian Federation	Germany	84 509
Turkey	Germany	81 404
Vietnam	USA	77 735
Bosnia and Herzegovina	Germany	75 678
Tajikistan	Russian Federation	72 556
Azerbaijan	Russian Federation	69 943
Romania	Germany	67 552
Kyrgyzstan	Russian Federation	65 385
Philippines	USA	61 022
Russian Federation	Belarus	57 520
Georgia	Russian Federation	54 247
Germany	Romania	52 367

The ranking by centrality indices is presented in Table 2.10.

The ranking by weighted in-degree centrality represents not only Germany and the USA as traditional immigration countries of the previous decades, but also the Russian Federation and Ukraine. The highest emigration rate (or weighted out-degree centrality) have the Russian Federation, Germany, Yugoslavia and Kazakhstan. Germany, the USA, the Russian Federation and Ukraine had consequently the largest gross migration. The weighted degree difference centrality has ranked the USA, Germany, the Russian Federation and Ukraine as with the largest net migration.

Countries with the highest closeness centrality index are Canada, the USA, Germany and New Zealand. Canada and New Zealand are represented because of their migration flows to and from the USA and Australia, correspondingly.

Both PageRank and eigenvector centralities rank countries based not only on their migration flows, but also account for their direct neighbors. The difference between PageRank and eigenvector centrality is that eigenvector additionally accounts for the number of migrants in migration flows. PageRank and Eigenvector results for 1992 differ considerably. PageRank apart from the USA and Germany has ranked the UK, Australia and Canada, which have had outflows of migrants to the USA of 40 000, 11 150 and 15 205, correspondingly. Eigenvector centrality has presented Germany, Ukraine, the Russian Federation, Poland and Yugoslavia among top five countries. Poland and Yugoslavia had the outflow of migrants to Germany of 143 709 and 267 thousand, correspondingly.

Overall, the results estimated by classical centrality indices for the 1990s provide the ranking of countries that complies the same idea, as in the previous decades. The two main groups of countries are always introduced: 1) with huge migration inflows and outflows (Germany, the USA and the Russian Federation in 1992), and 2) countries directly connected through migration flows with previous group (Canada, the UK, Kazakhstan and Yugoslavia).

Table 2.10. Rankings by centrality indices for 1992

Country	WInDeg	WOutDeg	WDeg	WDegDiff	Clos	PageRank	EigenVec	SRIC	LRIC (SUM)	LRIC (MAX)	LRIC (MAXMIN)
Germany	1	2	1	2	2	1	1	3	6	13	13
USA	2	13	3	1	3	2	7	12	15	15	18
Russian Federation	3	1	2	3	18	10	3	2	2	3	3
Ukraine	4	5	4	5	19	21	2	5	3	2	2
Canada	5	39	10	4	1	5	25	57	37	50	42
Australia	6	9	8	7	11	4	27	7	19	28	25
UK	7	8	7	24	22	3	11	8	10	16	23
Switzerland	8	23	12	6	6	9	13	64	55	56	64
Poland	9	7	9	208	16	6	4	6	9	9	10
Netherlands	10	28	15	8	7	8	20	37	43	45	46
Yugoslavia (former)	11	3	5	214	10	7	5	212	212	212	212
Kazakhstan	12	4	6	213	21	35	6	4	1	1	1
Italy	13	25	16	10	27	11	12	29	22	27	24
Belarus	14	30	22	13	57	48	9	39	27	17	21
Croatia	15	48	26	9	26	23	18	26	24	23	19
Romania	16	17	17	194	24	15	8	34	11	12	12
Turkey	17	11	13	205	23	14	10	11	5	6	6
Sweden	18	50	32	12	8	18	40	18	59	73	73
New Zealand	20	31	25	21	9	12	51	17	29	29	26
Denmark	22	46	33	16	4	13	36	53	64	71	66
Uzbekistan	24	10	14	210	55	64	15	13	4	4	4
Norway	28	66	47	17	5	22	56	25	74	86	83
Azerbaijan	34	19	24	201	132	72	23	30	14	10	9
Bosnia and Herzegovina	47	12	19	211	28	43	41	14	7	7	8
Tajikistan	50	18	27	206	136	106	37	28	13	8	7
Philippines	62	16	23	209	29	49	64	9	20	22	22
Vietnam	64	14	20	212	25	58	47	10	17	19	17
Mexico	82	6	11	215	13	77	76	1	8	5	5

Estimation of the Short and Long-Range Interactions Centrality indices provides different from the previous one list of countries.

SRIC ranking presents mostly the emigration countries. This fact is also confirmed by the Goodman-Kruskal correlation coefficient (Table 2.11) between SRIC and weighted out-degree indices (0.8). LRIC indices rank countries of the fSU in top four: Kazakhstan, Uzbekistan, The Russian Federation and Ukraine. Other emigration countries are also considered by these indices.

Comparison of the classical centrality indices and the SRIC and LRIC results lead to a conclusion that Short-Range and Long-Range Interactions Centrality indices are highly related to the weighted out-degree and weighted degree centralities ( 0.8) and compared to PageRank and eigenvector, they outline additionally countries with intense emigration flows, not only the countries connected to the attractive migrants' destinations.

Table 2.11. Goodman, Kruskal -coefficient for 1992

	SRIC	LRIC (SUM)	LRIC (MAX)	LRIC (MAXMIN)
WInDeg	0.669	0.719	0.698	0.714
WOutDeg	0.818	0.84	0.832	0.793
WDeg	0.812	0.848	0.823	0.797
WDegDiff	-0.426	-0.421	-0.427	-0.386
Clos	0.642	0.643	0.624	0.607
PageRank	0.638	0.673	0.647	0.675
Eigenvec	0.64	0.708	0.694	0.681
SRIC	1	0.873	0.864	0.841
LRIC (SUM)		1	0.957	0.924
LRIC (MAX)			1	0.919
LRIC (MAXMIN)				1

The last period of the international migration presented in our database is from 2000 to 2013. The major international migration flows occurred between the following groups of countries. First, the migration flow from Mexico, the Philippines and Vietnam to the United States were still of considerable level. Second, new Asian countries, India and China, appeared among labor force suppliers for the United States. The next destination of migrants from the developing countries was Spain. The immigrants from Ecuador, Morocco, Colombia and Argentina were moving to Spain till the beginning of the economic crisis in 2008. After 2008 Spain is no longer attractive for immigrants due to the high level of unemployment and becomes an emigration country [24]. Flows between the fSU countries were diminishing after 2007 and migration from the Russian Federation and Kazakhstan to Germany was decreasing accordingly. According to Eurostat statistics [45] Greece was one of the countries that experienced the highest growth in number of international migrants in recent time. Also, it was among the countries highly involved in migration in 1972 and was ranked by LRIC indices in top 10 countries. However, since 1998 Greece is no longer presented in the databases, that is why the rankings by centrality measures do not contain this country.

These processes in international migration in the last decade lead to the development of the following migration flows over 50000 (Table 2.12) in 2013.

Table 2.12. Migration flows over 50 000 in 2013

Origin	Destination	Migration flow
Mexico	USA	135028
China	USA	71798
Spain	Romania	70055
India	USA	68458
Romania	Italy	59347
Philippines	USA	54446

The considerable reduction in the number of international migration flows over 50 000 is observed compared to 1992. Centrality indices reflect these changes noticeably (Table 2.13).

From the results for weighted in-degree centrality we can conclude that the highest number of immigrants were received by the USA, Italy and the United Kingdom. According to the ranking by weighted out-degree, Spain, India and China had the highest migrant out-flow. Weighted degree ranking highlights the USA, Spain, Italy and the United Kingdom, which had the greatest gross migration rate. The weighted degree difference or the highest net migration flow was in the United States, Canada, the United Kingdom and Italy.

Different results can be obtained from the estimation of the level of closeness: the USA is still the first, however, Mexico, Netherlands, Spain and Switzerland are presented. These countries had intense migration in-flows (the USA) or out-flows (Spain) itself, or had migration flows to or from the countries with intense migration [36]. Mexico-US migration route was established historically, and now Mexicans are accounted for 28% of foreign-born population in the USA [46]. Netherlands and Switzerland were connected through migration flows to Italy, which was the second immigration country after the USA.

Eigenvector and PageRank highlight the “rich-club” group of countries: the United States, Italy, the United Kingdom and Spain. These countries are involved in the process of migration more than others and in addition had flows between each other. In this case eigenvector and PageRank centralities can show how “mobile” is the population of countries.

Table 2.13. Rankings by centrality indices for 2013

Country	WInDeg	WOutDeg	WDeg	WDegDiff	Clos	PageRank	EigenVec	SRIC	LRIC (SUM)	LRIC (MAX)	LRIC (MAXMIN)
USA	1	19	1	1	1	1	2	22	6	10	10
Italy	2	5	3	4	6	6	4	11	10	11	16
UK	3	10	4	3	30	3	1	9	9	4	7
Canada	4	44	5	2	10	7	12	74	37	43	30
Spain	5	1	2	215	3	2	3	1	1	1	1
Switzerland	6	12	7	6	5	5	6	35	44	54	80
Netherlands	7	8	8	10	4	8	11	17	14	23	27
Sweden	8	21	15	5	9	11	19	15	30	38	35
Belgium	9	14	10	9	7	12	9	23	19	28	45
Romania	10	6	6	198	14	17	5	2	2	2	2
Germany	11	11	9	23	37	10	7	12	4	8	9
New Zealand	12	16	13	14	8	4	14	5	23	15	15
France	13	9	12	192	36	15	8	7	3	5	5
Norway	14	52	23	7	11	16	23	32	45	49	24
Australia	15	31	22	8	33	9	20	18	21	25	21
Morocco	18	17	20	166	31	21	10	8	7	7	6
Poland	23	13	18	210	44	20	28	4	8	6	4
India	32	2	11	214	24	26	32	3	5	3	3
Mexico	45	4	14	212	2	56	40	49	40	35	65
Philippines	53	7	19	211	28	48	51	16	16	12	11
China	73	3	16	213	23	55	90	6	11	9	8
Syrian Arab Republic	133	37	49	200	61	137	137	10	28	27	22

Ranking by classic centrality indices has provided us with the information about countries with the highest in- and out-flows of migrants, net migration flow, level of closeness to huge migration flows and countries most involved in migration process. Short-Range and Long-Range Interactions Centralities can help us to explore the international migration network from the different perspective.

Spain, Romania, India and Poland have the highest ranks according to the index of short-range interactions. These results are highly related to the weighted out-degree. Additionally, SRIC accounts for the first-order indirect interactions and the population of destination country. That is why there was a little change in the order of countries with intense emigration flows.

Three of LRIC indices show almost similar results: Spain, Romania, France, Germany, Poland and India are at the top of rankings. Spain has the highest emigration rate. Romania, India and France have the migration flows to countries with huge population and intense migration flows. There was a huge flow from India to the USA, the USA has large population and is a popular country of migrants' destination (Migration Policy Institute). France is presented in ranking by LRIC indices, because it has migration flows to Spain (10548) and to the United Kingdom (24313). Romania also had migration flows to the United Kingdom. Poland did not appear among the countries with highest emigration rate (weighted out-degree), however, it had migration flow of almost 10000 migrants to Norway with population of around 5 million people. The share of this migrant inflow (0,2%) exceeded 0,1% of the population of Norway. This result is important to be considered as when migration flow is more than level expected by the destination country, it can lead to negative consequences for both migrants and the population of destination country.

The results introduced by classic centralities and SRIC, LRIC indices both outline the emigration countries. However, SRIC and LRIC indices introduce additionally the emigration countries with considerable for the population of destination country share of migrants (Poland).

Table 2.14. Goodman, Kruskal -coefficient for 2013

	SRIC	LRIC (SUM)	LRIC (MAX)	LRIC (MAXMIN)
WInDeg	0,716	0,746	0,716	0,742
WOutDeg	0,839	0,793	0,774	0,742
WDeg	0,831	0,798	0,798	0,742
WDegDiff	-0,414	-0,359	-0,359	-0,341
Clos	0,704	0,69	0,69	0,642
PageRank	0,716	0,76	0,76	0,714
Eigenvec	0,705	0,729	0,704	0,676
SRIC	1	0,845	0,845	0,799
LRIC (SUM)		1	0,934	0,885
LRIC (MAX)			1	0,9
LRIC (MAXMIN)				1

The main goal of the analysis of each decade provided above is to represent major migration flows in terms of the network analysis by introducing the ranking of countries based on the centrality indices. Overall, the analysis of classic and SRIC, LRIC centralities by decades has introduced the following outcomes. First, classic centrality indices analysis has outlined the occurrence and development of the major international migration flows in each decade. Second, SRIC and LRIC have outlined the influence of these flows on the population on destination countries and changes in interconnections in international migration network.

Estimation of classic centrality indices is the one of the possible ways to analyze countries' influence in the network through migration flows. New indices go a step further and allow to consider indirect connections of countries in the international migration network and a node attribute - the population of destination country. This idea is implied through Short-Range and Long-Range Interactions Centralities.

The analysis has been applied to annual data on migration flows, the results of the estimations have been compared for each decade. Our methodology has outlined not only the countries with large number of immigrants or emigrants, but also the countries with migrant outflows considerable for the population of destination country and emigration to the popular destination countries. These results are important in order to provide countries highly involved in the process of international migration with relevant migration policy.

Chapter 3. Econometric model of international migration

This chapter deals with the analysis of factors influencing bilateral migrant stock. After migration flows have been studied by network models, it is interesting to find the main determinants of these flows formation. The influence of traditional economic and demographic factors (GDP and population) alongside with distance was extensively studied in literature by various econometric models. However, the role of conflicts and level of education received less attention of researchers. In this Chapter the role of education level and magnitude of conflicts at origin will be examined by panel data models. First, the databases on international migrant stock and explanatory variables are provided. Second, the main panel data model specifications are described. Third, the results of model are analyzed.

3.1 Data for econometrics

The estimation of gravity model is based on the data on international migrant stock. The data for explanatory variables include the following groups of indicators: GDP per capita, intercountry distances, the estimates of education level and conflicts.

The information about migrant population was taken from the database Trends in International Migrant Stock: The 2015 Revision [42]. The database contains the estimates of total migrant stock by area of origin at the midyear (1^st of July) in 1990, 1995, 2000, 2005, 2010, 2015 for 232 countries. In most cases the data were obtained from the population censuses.

The migrant stock refers to the foreign-born population, if the information was available (180 countries). In other cases, the data on the number of foreign citizens were used. The application of data based on citizenship has certain shortcomings. For example, persons born abroad, but naturalized in the destination country are not included in international migration statistics. On the contrary, persons born in the country of residence who have foreign citizenship are recognized as international migrants. The data based on foreign citizenship were available for 46 countries.

International migrant stock included the number of refugees by country of origin. Population censuses do not always estimate the number of refugees in the country. If refugees have a legal status and are allowed to integrate in the country, they commonly are included to the total population of the country. However, in many countries refugees live in specialized camps or other isolated territories. In this case population registers do not cover them. Moreover, population census does not reflect to the sharp inflow of refugees, as it may not necessary be carried out right after such events. Consequently, for the 70 countries with large number of refugees the data in the database was obtained from the international organizations: Office of the United Nations High Commissioner for Refugees (UNHCR) and the United Nations Relief and Works Agency for Palestine Refugees in the Near East (UNWRA).

The database for gravity model analysis developed by the Institute for Research on the International Economy (CEPII) is considered. The Geodist database is constructed by (Mayer et al., 2011) for the analysis of international trade flows. Geodist contains the country-pair data for 225 countries about the common border, language, colonial past and bilateral distances (in km) measured by different approaches. There are 8 dummy variables for gravity model and 4 variables for distances. Two approaches are applied for distance estimation. Simple distance is measured by the geodesic formula of the shortest path between the two points on earth using its latitudes and longitudes coordinates (the great circle formula). This distance is measured between the capitals of countries and the most populated cities (for 212 countries these cities are the same). The database also provides the interstate distances, which account for the internal distances between the cities in the countries. The distance is calculated as a sum of distances between the regions of two countries weighted by the share of population of each region in the total country population.

Data on GDP per capita based on purchasing power parity in current international dollars are considered. Data is available for 217 countries from 1990 to 2014 [48].

The data on the level of education is explored with Education Index, which is included in the Human Development Index [43]. The estimation of Education Index is based on the mean years of schooling and expected years of schooling. Data is available for 187 countries from 1980 to 2013.

Another indicator of interest was the intensity of conflicts in the origin country. The “Major Episodes of Political Violence, 1946-2015” (MEPV) database provided by Center for Systemic Peace (CSP) is considered [44]. The database includes 179 countries with population of more than 500000 in 2015 and period from 1946 to 2013. Also, the database includes only conflicts, where the number of deaths was greater than 500. The database evaluates the intensity of conflict for each country in every period of conflict. The magnitude of conflict is measured by the scale from 1 to 10. Conflicts are classified by 7 categories: international violence, international war, international independence war, civil violence, civil war, ethnic violence and ethnic war. The index, which includes sum by all categories of conflict is taken for the analysis. Each conflict refers to one of the ten categories of intensity, which is evaluated in the integrated approach by experts. The methodology is based on different criteria including the number of deaths, injuries, population displacement and others.

3.2 Model Specification

The estimation of the regression equation is based on the theoretical model specification (1.4). The standard procedure of estimation of the gravity equation is to take logarithms of all the variables and move to log-linear model specification, which can be estimated with ordinary least squares regression. The following econometric model is chosen for the estimation:



(3.1)

where represents migrant stock from country i in country j, where and , .

Explanatory variables are the following. and denote the logarithm of GDP per capita of origin and destination country respectively, lagged by 1 year. The basic hypothesis of gravity model assumes the positive relationship between GDP per capita of origin and destination countries and migrant stock. In our model GDP per capita introduces the economic prospects for migrants in home and host countries. GDP of origin country represents the main pull factor of international migration. However, as literature shows, this type of models is related with reverse causality problem, i.e. the increase of GDP at destination can be caused by the increase of migrant stock. In other words, migrant stock in current period and GDP are expected to be highly correlated. That is why in the estimation of models the values of GDP per capita are lagged by 1 year.

The distance variable () is presented with the distance between capital cities of origin and destination countries. Distance in our model represents the costs of migration. As it was shown by [2], distance is a significant factor influencing the movement of people between the Russian regions. It is interesting to explore this relationship between distance and migrant stock on the country level.

On the contrary, the size of existing stock of migrants in the destination country (diaspora) is a significant factor that reduces the costs of migration. As it was shown by [17, 19, 33] diaspora is the strong pull factor of migration alongside with per capita GDP at destination. Migrant stock from country i in country j lagged by 5 periods () is included in our model.