Quantifying the status of economies in international crop trade networks: An correlation structure analysis of various node-ranking metrics

TL;DR

This study analyzes international crop trade networks from 1986 to 2020, evaluating the importance of economies using multiple metrics and identifying key players to enhance food security and trade stability.

Contribution

It introduces a comprehensive correlation structure analysis of node importance metrics in international crop trade networks, integrating 20 metrics into a unified measure.

Findings

European economies significantly influence international crop trade networks

Poor crop-producing economies are key import players

High food production economies are crucial for exports

Abstract

International food trade is a growing complement to gaps in domestic food supply and demand, but it is vulnerable to disruptions due to some unforeseen shocks. This paper assembles the international crop trade networks using maize, rice, soybean, and wheat trade data sets from 1986 to 2020. We assess the importance of economies using multidimensional node importance metrics. We analyze the correlation structure of different node important metrics based on the random matrix theory and incorporate 20 metrics into a single metric. We find that some metrics have many similarities and dissimilarities, especially for metrics based on the same trade flow directions. We also find that European economies have a significant impact on the iCTNs. Additionally, economies with poor crop production play a major role in import trade, whereas economies with higher food production or smaller populations are crucial to export trade. Our findings have practical implications for identifying key economies in the international crop trade networks, preventing severe damage to the food trade system caused by trade disruptions in some economies, maintaining the stability of the food supply, and ensuring food security.

Tables

Table 1. Table 1: Top 10 economies identified based on different influence indicators in 2020. ID (in-degree centrality), OD (out-degree centrality), IS (in-strength), OS (out-strength), BC (betweenness centrality), EC (eigenvector centrality), KC (Katz centrality), CC (clustering centrality), IC (in-closeness centrality), OC (out-closeness centrality), PR (PageRank), HU (hubs), AU (authorities), CI (constraint index), IL (in-semi-local centrality), OL (out-semi-local centrality), MI (mutual information), IM (within-module), PT (participation coefficient), OM (outside-module).

Rank

ID

OD

IS

OS

BC

EC

KC

CC

IC

OC

PR

HU

AU

CI

IL

OL

MI

IM

PT

OM

Panel A: Maize

1

NLD

USA

MEX

USA

USA

NLD

HKG

HTI

USA

NLD

USA

GBR

USA

MEX

NLD

USA

USA

ARG

NPL

USA

2

GBR

ARG

JPN

ARG

ARE

DEU

USA

SLE

ARG

USA

ARG

IRL

BRA

JPN

FRA

ARG

ARG

USA

MAR

ARG

3

USA

BRA

VNM

BRA

ZAF

FRA

MYS

CUB

FRA

FRA

BRA

NLD

UKR

COL

DEU

FRA

BRA

ZAF

JPN

ZAF

4

FRA

FRA

CHN

UKR

FRA

GBR

KEN

ERI

BRA

ARE

FRA

DEU

ARG

CHN

GBR

BRA

FRA

BRA

MUS

ARE

5

DEU

ZAF

KOR

FRA

NLD

ESP

AUS

VUT

DEU

GBR

ZAF

NER

ZAF

KOR

ARE

DEU

NLD

FRA

NGA

BRA

6

ARE

UKR

EGY

HUN

TUR

DNK

ARE

LCA

ESP

DEU

UKR

BEN

RUS

VNM

USA

TUR

UKR

IND

HKG

FRA

7

CAN

TUR

ESP

ZAF

CAN

BEL

THA

FJI

TUR

CAN

TUR

TUR

BGR

CAN

ESP

ESP

ZAF

UKR

AUS

NLD

8

BEL

ESP

COL

BGR

GBR

ARE

PHL

LUX

UKR

JPN

IND

BEL

FRA

TWN

BEL

UKR

CAN

ESP

THA

IND

9

ITA

IND

IRN

RUS

IND

ITA

IDN

TJK

MEX

KWT

ESP

ESP

PRY

IRN

DNK

MEX

DEU

NLD

GUY

TUR

10

ESP

HUN

ITA

PRY

DEU

HUN

UGA

TKM

ITA

SGP

DEU

FRA

HUN

GTM

CAN

ITA

IND

TUR

NZL

DEU

Panel B: Rice

1

NLD

IND

MEX

USA

ARE

NLD

NLD

AFG

ITA

GBR

IND

ESP

USA

MEX

NLD

ITA

IND

IND

JPN

IND

2

GBR

ITA

NPL

BRA

ITA

DEU

DEU

ALB

NLD

NLD

ITA

NLD

BRA

COL

DEU

NLD

BRA

CHN

URY

CHN

3

DEU

BRA

VEN

IND

GBR

BEL

BEL

OMN

DEU

ITA

USA

DEU

PRY

VEN

BEL

DEU

USA

USA

GEO

NLD

4

ARE

NLD

BRA

URY

NLD

GBR

GBR

SYC

IND

ESP

CHN

PRT

URY

BRA

GBR

BEL

NLD

ARE

PAN

USA

5

BEL

USA

CRI

PRY

USA

POL

POL

DOM

BEL

ARE

BRA

GBR

GUY

HND

ITA

BRA

ITA

PAK

LBN

GBR

6

ITA

DEU

COL

GUY

BRA

ITA

ITA

SVK

BRA

DEU

PAK

BEL

RUS

NIC

POL

GBR

ARE

ZAF

LKA

ITA

7

ESP

CHN

TUR

KHM

IND

DNK

DNK

LUX

GBR

IRL

ARE

IRL

BGR

GTM

ISL

IND

CHN

NLD

MRT

THA

8

FRA

ARE

GTM

GRC

CHN

ISL

ISL

EST

ESP

FRA

GBR

TUR

GRC

CRI

IRL

ESP

ESP

ITA

GMB

ARE

9

ISL

TUR

HND

RUS

TUR

HUN

IRL

SVN

FRA

BEL

LKA

AUT

ITA

SLV

ESP

FRA

GBR

BRA

CYP

DEU

10

CHE

GBR

PAN

BGR

DEU

IRL

HUN

FIN

PRT

NOR

BEL

ITA

ESP

TUR

FRA

PRT

PAK

ESP

NGA

JPN

Panel C: Soybean

1

NLD

USA

CHN

BRA

USA

NLD

NLD

YEM

CAN

NLD

USA

ESP

BRA

CHN

NLD

CAN

USA

USA

SWE

USA

2

DEU

CAN

ARG

USA

CAN

DEU

DEU

HND

USA

FRA

CAN

PRT

USA

THA

DEU

USA

CAN

IND

JPN

CAN

3

FRA

CHN

NLD

ARG

IND

FRA

AUT

JAM

CHN

DEU

CHN

DEU

ARG

NLD

FRA

CHN

NLD

CAN

JOR

NLD

4

USA

BRA

EGY

PRY

CHN

ITA

HUN

PAN

BRA

ESP

BRA

FRA

URY

ESP

ITA

BRA

BRA

NLD

HRV

FRA

5

ESP

UKR

THA

CAN

NLD

ESP

POL

LBY

UKR

GBR

IND

CHN

RUS

TUR

ESP

UKR

DEU

UKR

CHN

DEU

6

ITA

FRA

MEX

URY

FRA

AUT

ITA

SUR

AUT

ITA

UKR

USA

CAN

MEX

GBR

AUT

CHN

DEU

MDA

CHN

7

GBR

DEU

DEU

UKR

ZAF

GBR

FRA

BRB

ARG

CAN

AUT

BEL

UKR

TWN

AUT

NLD

IND

FRA

CYP

GBR

8

IND

AUT

ESP

NLD

ARE

BEL

DNK

TTO

FRA

USA

ARG

NLD

PRY

PAK

BEL

FRA

FRA

AUT

IRN

ITA

9

CAN

NLD

JPN

RUS

DEU

POL

SVK

OMN

NLD

BEL

NLD

IND

ETH

JPN

POL

ARG

UKR

BRA

CRI

AUT

Panel D: Wheat

1

GBR

RUS

IDN

RUS

FRA

GBR

BOL

VEN

FRA

GBR

CAN

ZAF

RUS

TUR

GBR

FRA

USA

USA

ISR

CAN

2

NLD

CAN

TUR

CAN

USA

NLD

SYC

TJK

DEU

FRA

FRA

ITA

UKR

EGY

NLD

DEU

RUS

RUS

IRN

USA

3

DEU

FRA

EGY

USA

ARE

DEU

LKA

TKM

ITA

NLD

DEU

ESP

CAN

BGD

DEU

ITA

FRA

CAN

CYP

RUS

4

ITA

USA

CHN

FRA

CAN

ESP

AFG

SUR

POL

BEL

RUS

NLD

USA

IDN

ESP

POL

DEU

ITA

MLT

FRA

5

TUR

DEU

ITA

UKR

DEU

FRA

PHL

NPL

AUT

DEU

USA

TUR

FRA

CHN

FRA

AUT

CAN

UKR

ALB

DEU

6

BEL

UKR

DZA

ARG

GBR

ITA

HKG

LCA

CZE

MYS

UKR

DEU

ARG

PHL

ITA

CZE

UKR

DEU

EGY

ITA

7

ESP

POL

BRA

AUS

TWN

CHE

MDV

TTO

NLD

ARE

ITA

BEL

AUS

NGA

BEL

NLD

AUS

AUS

CZE

UKR

8

FRA

IND

BGD

DEU

RUS

BEL

NZL

DOM

LTU

KEN

POL

ARE

LTU

MAR

CHE

LTU

NLD

POL

TUR

TUR

9

CHE

ITA

NGA

KAZ

ZAF

AUT

GTM

BRB

HUN

ESP

IND

GBR

DEU

JPN

AUT

HUN

POL

BGR

IRL

KAZ

10

AUT

LTU

PHL

LTU

TUR

TUR

IND

FRO

RUS

THA

TUR

SGP

BGR

PAK

TUR

RUS

BEL

KAZ

MOZ

GBR

Table 2. Table 2: Top 10 economies identified based on composite node importance index (CNI).

Rank

Maize

Rice

Soybean

Wheat

${\mathrm{CNI}}^{\mathrm{in}}$

${\mathrm{CNI}}^{\mathrm{out}}$

${\mathrm{CNI}}^{\mathrm{third}}$

${\mathrm{CNI}}^{\mathrm{in}}$

${\mathrm{CNI}}^{\mathrm{out}}$

${\mathrm{CNI}}^{\mathrm{third}}$

${\mathrm{CNI}}^{\mathrm{in}}$

${\mathrm{CNI}}^{\mathrm{out}}$

${\mathrm{CNI}}^{\mathrm{third}}$

${\mathrm{CNI}}^{\mathrm{in}}$

${\mathrm{CNI}}^{\mathrm{out}}$

${\mathrm{CNI}}^{\mathrm{third}}$

1

NLD

USA

LSO

GBR

BRA

CZE

NLD

USA

MLT

NLD

FRA

AUT

2

GBR

ARG

TON

NLD

ITA

SVK

DEU

CAN

HUN

ITA

CAN

ESP

3

JPN

FRA

NER

ITA

USA

HUN

ESP

BRA

SVN

TUR

RUS

NLD

4

ESP

BRA

BRB

DEU

IND

EST

ITA

UKR

LVA

GBR

DEU

GRC

5

DEU

ZAF

MOZ

ESP

GBR

LUX

FRA

CHN

SVK

ESP

USA

MAR

6

ITA

UKR

NAM

BEL

NLD

FIN

GBR

NLD

EST

BEL

UKR

DEU

7

CAN

IND

USA

FRA

CHN

POL

BEL

AUT

IRL

DEU

POL

CZE

8

BEL

NLD

ARG

PRT

ESP

SVN

CHN

DEU

BLR

ARE

LTU

DZA

9

TUR

TUR

BRA

IRL

DEU

AUT

PRT

FRA

BIH

MAR

GBR

BIH

10

ARE

DEU

ZAF

ARE

PAK

BGR

CAN

ARG

CZE

ISR

IND

HUN

Table 3. Table A1: Correlation of the influence indicators in 2020.

	OD	ID	IS	OS	BC	EC	KC	CC	IC	OC	PR	HU	AU	CI	IL	OL	MI	IM	PT	OM
OD	1.000	0.698	0.571	0.916	0.898	0.626	-0.015	-0.241	0.602	0.942	0.488	0.906	0.465	0.276	0.644	0.937	0.101	0.740	0.372	0.758
ID	0.698	1.000	0.654	0.588	0.707	0.957	-0.110	-0.053	0.945	0.637	0.714	0.573	0.518	-0.041	0.971	0.661	0.036	0.673	0.449	0.746
IS	0.571	0.654	1.000	0.443	0.558	0.577	0.120	0.027	0.603	0.602	0.670	0.515	0.884	0.135	0.602	0.565	0.240	0.471	0.458	0.618
OS	0.916	0.588	0.443	1.000	0.800	0.548	-0.040	-0.196	0.508	0.850	0.408	0.943	0.343	0.361	0.563	0.857	0.172	0.677	0.294	0.639
BC	0.898	0.707	0.558	0.800	1.000	0.614	0.094	-0.372	0.642	0.867	0.590	0.779	0.454	0.218	0.638	0.841	0.155	0.730	0.392	0.792
EC	0.626	0.957	0.577	0.548	0.614	1.000	-0.268	0.037	0.956	0.554	0.617	0.521	0.442	-0.056	0.996	0.592	0.015	0.580	0.408	0.654
KC	-0.015	-0.110	0.120	-0.040	0.094	-0.268	1.000	-0.198	-0.084	0.019	0.190	0.000	0.259	0.121	-0.212	-0.080	0.003	0.120	0.299	0.192
CC	-0.241	-0.053	0.027	-0.196	-0.372	0.037	-0.198	1.000	-0.075	-0.211	-0.182	-0.137	0.024	0.181	0.019	-0.167	-0.386	-0.276	-0.056	-0.307
IC	0.602	0.945	0.603	0.508	0.642	0.956	-0.084	-0.075	1.000	0.542	0.687	0.482	0.515	-0.072	0.967	0.548	0.051	0.613	0.489	0.734
OC	0.942	0.637	0.602	0.850	0.867	0.554	0.019	-0.211	0.542	1.000	0.475	0.885	0.502	0.302	0.577	0.974	0.110	0.648	0.390	0.738
PR	0.488	0.714	0.670	0.408	0.590	0.617	0.190	-0.182	0.687	0.475	1.000	0.428	0.527	0.044	0.645	0.463	0.261	0.547	0.345	0.631
HU	0.906	0.573	0.515	0.943	0.779	0.521	0.000	-0.137	0.482	0.885	0.428	1.000	0.442	0.360	0.538	0.885	0.166	0.636	0.295	0.643
AU	0.465	0.518	0.884	0.343	0.454	0.442	0.259	0.024	0.515	0.502	0.527	0.442	1.000	0.132	0.472	0.429	0.178	0.459	0.431	0.610
CI	0.276	-0.041	0.135	0.361	0.218	-0.056	0.121	0.181	-0.072	0.302	0.044	0.360	0.132	1.000	-0.045	0.240	-0.256	0.034	0.158	0.017
IL	0.644	0.971	0.602	0.563	0.638	0.996	-0.212	0.019	0.967	0.577	0.645	0.538	0.472	-0.045	1.000	0.610	0.022	0.610	0.427	0.686
OL	0.937	0.661	0.565	0.857	0.841	0.592	-0.080	-0.167	0.548	0.974	0.463	0.885	0.429	0.240	0.610	1.000	0.105	0.637	0.335	0.694
MI	0.101	0.036	0.240	0.172	0.155	0.015	0.003	-0.386	0.051	0.110	0.261	0.166	0.178	-0.256	0.022	0.105	1.000	0.208	-0.052	0.190
IM	0.740	0.673	0.471	0.677	0.730	0.580	0.120	-0.276	0.613	0.648	0.547	0.636	0.459	0.034	0.610	0.637	0.208	1.000	0.165	0.756
PT	0.372	0.449	0.458	0.294	0.392	0.408	0.299	-0.056	0.489	0.390	0.345	0.295	0.431	0.158	0.427	0.335	-0.052	0.165	1.000	0.548
OM	0.758	0.746	0.618	0.639	0.792	0.654	0.192	-0.307	0.734	0.738	0.631	0.643	0.610	0.017	0.686	0.694	0.190	0.756	0.548	1.000

Table 4. Table A2: Correlation of the influence indicators in 2020.

	OD	ID	IS	OS	BC	EC	KC	CC	IC	OC	PR	HU	AU	CI	IL	OL	MI	IM	PT	OM
OD	1.000	0.402	0.221	0.932	0.904	0.347	0.350	0.043	0.348	0.964	0.378	0.912	0.157	0.666	0.371	0.888	0.171	0.615	0.238	0.574
ID	0.402	1.000	0.611	0.302	0.455	0.763	0.773	0.448	0.831	0.355	0.673	0.318	0.544	0.067	0.799	0.417	-0.038	0.603	0.496	0.651
IS	0.221	0.611	1.000	0.206	0.252	0.300	0.311	0.218	0.380	0.194	0.707	0.236	0.595	-0.044	0.355	0.189	0.429	0.480	0.237	0.307
OS	0.932	0.302	0.206	1.000	0.839	0.260	0.260	0.028	0.258	0.898	0.305	0.924	0.140	0.746	0.279	0.780	0.253	0.565	0.151	0.440
BC	0.904	0.455	0.252	0.839	1.000	0.374	0.377	-0.039	0.395	0.891	0.426	0.810	0.178	0.588	0.398	0.818	0.149	0.581	0.284	0.614
EC	0.347	0.763	0.300	0.260	0.374	1.000	0.999	0.482	0.958	0.308	0.447	0.275	0.396	0.132	0.990	0.430	-0.150	0.357	0.416	0.496
KC	0.350	0.773	0.311	0.260	0.377	0.999	1.000	0.482	0.961	0.310	0.455	0.276	0.402	0.129	0.992	0.434	-0.148	0.364	0.422	0.502
CC	0.043	0.448	0.218	0.028	-0.039	0.482	0.482	1.000	0.425	0.031	0.182	0.090	0.367	0.116	0.485	0.099	-0.283	0.127	0.179	0.062
IC	0.348	0.831	0.380	0.258	0.395	0.958	0.961	0.425	1.000	0.306	0.495	0.264	0.399	0.096	0.968	0.394	-0.116	0.442	0.447	0.573
OC	0.964	0.355	0.194	0.898	0.891	0.308	0.310	0.031	0.306	1.000	0.335	0.913	0.144	0.689	0.335	0.910	0.159	0.569	0.225	0.538
PR	0.378	0.673	0.707	0.305	0.426	0.447	0.455	0.182	0.495	0.335	1.000	0.322	0.267	0.090	0.495	0.369	0.341	0.510	0.237	0.405
HU	0.912	0.318	0.236	0.924	0.810	0.275	0.276	0.090	0.264	0.913	0.322	1.000	0.209	0.713	0.299	0.827	0.170	0.548	0.157	0.427
AU	0.157	0.544	0.595	0.140	0.178	0.396	0.402	0.367	0.399	0.144	0.267	0.209	1.000	-0.075	0.426	0.210	0.050	0.201	0.372	0.280
CI	0.666	0.067	-0.044	0.746	0.588	0.132	0.129	0.116	0.096	0.689	0.090	0.713	-0.075	1.000	0.140	0.530	-0.130	0.231	0.012	0.097
IL	0.371	0.799	0.355	0.279	0.398	0.990	0.992	0.485	0.968	0.335	0.495	0.299	0.426	0.140	1.000	0.446	-0.133	0.391	0.444	0.525
OL	0.888	0.417	0.189	0.780	0.818	0.430	0.434	0.099	0.394	0.910	0.369	0.827	0.210	0.530	0.446	1.000	0.124	0.537	0.176	0.517
MI	0.171	-0.038	0.429	0.253	0.149	-0.150	-0.148	-0.283	-0.116	0.159	0.341	0.170	0.050	-0.130	-0.133	0.124	1.000	0.235	-0.175	0.019
IM	0.615	0.603	0.480	0.565	0.581	0.357	0.364	0.127	0.442	0.569	0.510	0.548	0.201	0.231	0.391	0.537	0.235	1.000	0.104	0.648
PT	0.238	0.496	0.237	0.151	0.284	0.416	0.422	0.179	0.447	0.225	0.237	0.157	0.372	0.012	0.444	0.176	-0.175	0.104	1.000	0.620
OM	0.574	0.651	0.307	0.440	0.614	0.496	0.502	0.062	0.573	0.538	0.405	0.427	0.280	0.097	0.525	0.517	0.019	0.648	0.620	1.000

Table 5. Table A3: Correlation of the influence indicators in 2020.

	OD	ID	IS	OS	BC	EC	KC	CC	IC	OC	PR	HU	AU	CI	IL	OL	MI	IM	PT	OM
OD	1.000	0.609	0.575	0.913	0.880	0.506	0.162	-0.086	0.504	0.941	0.556	0.939	0.467	0.449	0.527	0.963	0.056	0.693	0.445	0.753
ID	0.609	1.000	0.719	0.501	0.669	0.914	0.359	0.225	0.927	0.511	0.805	0.523	0.618	-0.019	0.928	0.579	-0.017	0.683	0.586	0.807
IS	0.575	0.719	1.000	0.469	0.631	0.637	0.076	0.188	0.678	0.548	0.708	0.499	0.894	0.092	0.663	0.550	0.148	0.566	0.585	0.657
OS	0.913	0.501	0.469	1.000	0.775	0.402	0.168	-0.034	0.365	0.874	0.460	0.945	0.344	0.530	0.413	0.887	0.144	0.658	0.364	0.641
BC	0.880	0.669	0.631	0.775	1.000	0.537	0.097	-0.146	0.568	0.829	0.650	0.791	0.507	0.343	0.560	0.831	0.139	0.664	0.438	0.746
EC	0.506	0.914	0.637	0.402	0.537	1.000	0.581	0.304	0.951	0.404	0.688	0.456	0.563	-0.052	0.991	0.506	-0.128	0.553	0.550	0.739
KC	0.162	0.359	0.076	0.168	0.097	0.581	1.000	0.051	0.415	0.073	0.159	0.191	0.022	-0.077	0.515	0.188	-0.022	0.146	0.128	0.295
CC	-0.086	0.225	0.188	-0.034	-0.146	0.304	0.051	1.000	0.233	-0.096	0.158	-0.014	0.168	0.083	0.324	-0.046	-0.423	0.000	0.308	0.043
IC	0.504	0.927	0.678	0.365	0.568	0.951	0.415	0.233	1.000	0.408	0.718	0.415	0.640	-0.113	0.970	0.487	-0.076	0.580	0.554	0.766
OC	0.941	0.511	0.548	0.874	0.829	0.404	0.073	-0.096	0.408	1.000	0.485	0.903	0.477	0.503	0.427	0.966	0.019	0.660	0.403	0.665
PR	0.556	0.805	0.708	0.460	0.650	0.688	0.159	0.158	0.718	0.485	1.000	0.467	0.571	0.082	0.712	0.518	0.124	0.563	0.427	0.660
HU	0.939	0.523	0.499	0.945	0.791	0.456	0.191	-0.014	0.415	0.903	0.467	1.000	0.390	0.513	0.468	0.928	0.020	0.665	0.418	0.679
AU	0.467	0.618	0.894	0.344	0.507	0.563	0.022	0.168	0.640	0.477	0.571	0.390	1.000	0.059	0.604	0.453	0.044	0.481	0.529	0.532
CI	0.449	-0.019	0.092	0.530	0.343	-0.052	-0.077	0.083	-0.113	0.503	0.082	0.513	0.059	1.000	-0.038	0.453	-0.219	0.068	0.151	0.083
IL	0.527	0.928	0.663	0.413	0.560	0.991	0.515	0.324	0.970	0.427	0.712	0.468	0.604	-0.038	1.000	0.525	-0.129	0.574	0.565	0.762
OL	0.963	0.579	0.550	0.887	0.831	0.506	0.188	-0.046	0.487	0.966	0.518	0.928	0.453	0.453	0.525	1.000	-0.002	0.673	0.432	0.723
MI	0.056	-0.017	0.148	0.144	0.139	-0.128	-0.022	-0.423	-0.076	0.019	0.124	0.020	0.044	-0.219	-0.129	-0.002	1.000	0.167	-0.165	0.095
IM	0.693	0.683	0.566	0.658	0.664	0.553	0.146	0.000	0.580	0.660	0.563	0.665	0.481	0.068	0.574	0.673	0.167	1.000	0.195	0.707
PT	0.445	0.586	0.585	0.364	0.438	0.550	0.128	0.308	0.554	0.403	0.427	0.418	0.529	0.151	0.565	0.432	-0.165	0.195	1.000	0.593
OM	0.753	0.807	0.657	0.641	0.746	0.739	0.295	0.043	0.766	0.665	0.660	0.679	0.532	0.083	0.762	0.723	0.095	0.707	0.593	1.000

Table 6. Table A4: Correlation of the influence indicators in 2020.

	OD	ID	IS	OS	BC	EC	KC	CC	IC	OC	PR	HU	AU	CI	IL	OL	MI	IM	PT	OM
OD	1.000	0.598	0.292	0.949	0.859	0.613	-0.508	-0.142	0.517	0.951	0.363	0.937	0.151	0.606	0.614	0.929	0.078	0.692	0.394	0.710
ID	0.598	1.000	0.550	0.545	0.647	0.971	-0.763	0.111	0.941	0.584	0.697	0.517	0.452	0.207	0.975	0.587	0.133	0.653	0.582	0.814
IS	0.292	0.550	1.000	0.219	0.319	0.489	-0.331	0.119	0.501	0.277	0.613	0.198	0.903	0.201	0.493	0.269	0.232	0.350	0.321	0.417
OS	0.949	0.545	0.219	1.000	0.809	0.563	-0.483	-0.119	0.464	0.906	0.328	0.985	0.074	0.644	0.563	0.879	0.104	0.639	0.344	0.629
BC	0.859	0.647	0.319	0.809	1.000	0.618	-0.449	-0.247	0.589	0.840	0.436	0.805	0.205	0.449	0.624	0.827	0.178	0.630	0.366	0.673
EC	0.613	0.971	0.489	0.563	0.618	1.000	-0.874	0.197	0.932	0.605	0.631	0.540	0.381	0.216	0.998	0.613	0.077	0.637	0.593	0.815
KC	-0.508	-0.763	-0.331	-0.483	-0.449	-0.874	1.000	-0.287	-0.734	-0.510	-0.405	-0.465	-0.222	-0.159	-0.857	-0.528	0.013	-0.430	-0.543	-0.645
CC	-0.142	0.111	0.119	-0.119	-0.247	0.197	-0.287	1.000	0.100	-0.091	0.056	-0.132	0.075	0.145	0.191	-0.067	-0.370	-0.138	0.179	-0.074
IC	0.517	0.941	0.501	0.464	0.589	0.932	-0.734	0.100	1.000	0.494	0.693	0.448	0.427	0.183	0.941	0.484	0.110	0.608	0.539	0.784
OC	0.951	0.584	0.277	0.906	0.840	0.605	-0.510	-0.091	0.494	1.000	0.339	0.913	0.139	0.608	0.606	0.973	0.028	0.641	0.418	0.692
PR	0.363	0.697	0.613	0.328	0.436	0.631	-0.405	0.056	0.693	0.339	1.000	0.312	0.476	0.163	0.643	0.346	0.311	0.438	0.328	0.496
HU	0.937	0.517	0.198	0.985	0.805	0.540	-0.465	-0.132	0.448	0.913	0.312	1.000	0.056	0.651	0.539	0.886	0.095	0.630	0.347	0.627
AU	0.151	0.452	0.903	0.074	0.205	0.381	-0.222	0.075	0.427	0.139	0.476	0.056	1.000	0.118	0.384	0.115	0.123	0.326	0.338	0.388
CI	0.606	0.207	0.201	0.644	0.449	0.216	-0.159	0.145	0.183	0.608	0.163	0.651	0.118	1.000	0.221	0.569	-0.373	0.234	0.227	0.210
IL	0.614	0.975	0.493	0.563	0.624	0.998	-0.857	0.191	0.941	0.606	0.643	0.539	0.384	0.221	1.000	0.612	0.083	0.643	0.589	0.820
OL	0.929	0.587	0.269	0.879	0.827	0.613	-0.528	-0.067	0.484	0.973	0.346	0.886	0.115	0.569	0.612	1.000	0.041	0.624	0.384	0.670
MI	0.078	0.133	0.232	0.104	0.178	0.077	0.013	-0.370	0.110	0.028	0.311	0.095	0.123	-0.373	0.083	0.041	1.000	0.227	-0.100	0.146
IM	0.692	0.653	0.350	0.639	0.630	0.637	-0.430	-0.138	0.608	0.641	0.438	0.630	0.326	0.234	0.643	0.624	0.227	1.000	0.323	0.801
PT	0.394	0.582	0.321	0.344	0.366	0.593	-0.543	0.179	0.539	0.418	0.328	0.347	0.338	0.227	0.589	0.384	-0.100	0.323	1.000	0.663
OM	0.710	0.814	0.417	0.629	0.673	0.815	-0.645	-0.074	0.784	0.692	0.496	0.627	0.388	0.210	0.820	0.670	0.146	0.801	0.663	1.000

Equations

$$D{C}_i=\frac{k_i}{N_{\mathcal{V}}-1}.$$

$$D{C}_i^{\mathrm{in}}=\frac{k_i^{\mathrm{in}}}{N_{\mathcal{V}}-1},$$

$$D{C}_i^{\mathrm{out}}=\frac{k_i^{\mathrm{out}}}{N_{\mathcal{V}}-1}.$$

$$L{C}_i=\sum _{u\in \tau (i)}\sum _{v\in \tau (u)}{k}_v,$$

$$I{L}_i=\sum _{u\in {\tau }^{\mathrm{in}}(i)}\sum _{v\in {\tau }^{\mathrm{in}}(u)}{k}_v^{\mathrm{in}},$$

$$O{L}_i=\sum _{u\in {\tau }^{\mathrm{out}}(i)}\sum _{v\in {\tau }^{\mathrm{out}}(u)}{k}_v^{\mathrm{out}},$$

$$\text{centering}\mathbf{A}\cdot \mathbf{e}$$

$$e_i$$

$$E{C}_i=e_i,$$

$$K{C}_i=\sum _{k=1}^{\infty }\sum _{j=1}^{N_{\mathcal{V}}}{\alpha }^k{a}_{ij}^k,$$

$$C{C}_i=\frac{1}{k_i(k_i-1)-2k_i^R}\sum _{sr}({\hat{w}}_{is}{\hat{w}}_{sr}{\hat{w}}_{ri}{)}^{1/3},$$

$$B{C}_i=\sum _{p,q\in \mathcal{V},p\ne q\ne i}\frac{h_{pq}^i}{g_{pq}},$$

$$I{C}_i=\frac{D_i^{\mathrm{in}}}{N_{\mathcal{V}}-1}\frac{D_i^{\mathrm{in}}}{{\sum }_{j=1}^{D_i^{\mathrm{in}}}{d}_{ji}},$$

$$O{C}_i=\frac{D_i^{\mathrm{out}}}{N_{\mathcal{V}}-1}\frac{D_i^{\mathrm{out}}}{{\sum }_{j=1}^{D_i^{\mathrm{out}}}{d}_{ij}},$$

$$I{M}_i=\frac{{\kappa }_{i,m_i}-{\overline{\kappa }}_{m_i}}{{\sigma }_{{\kappa }_{m_i}}},$$

$$P{T}_i=1-\sum _{m=1}^{N_M}{\left(\frac{{\kappa }_{i,m_i}}{k_i}\right)}^2,$$

$$O{M}_i=\frac{{\kappa }_{i,m_i}^{{}^{\prime }}-{\overline{\kappa }}_{m_i}^{{}^{\prime }}}{{\sigma }_{{\kappa }_{m_i}}},$$

$$I_{ij}=\{\begin{array}{ll}\ln k_i-\ln k_j,& j\in {\mathcal{V}}_i\\ 0,& \text{ others }\end{array}.$$

$$M{I}_i=\sum _{j=1}^N{I}_{ij},$$

$$p_i^{\mathrm{out}}=\frac{w_{ij}}{s_i^{\mathrm{out}}}.$$

$$p_j^{\mathrm{in}}=\frac{w_{ij}}{s_j^{\mathrm{in}}}.$$

$$M{I}_{ij}=\{\begin{array}{ll}\ln \frac{s_i^{\mathrm{out}}}{w_{ij}}-\ln \frac{s_j^{\mathrm{in}}}{w_{ij}},& j\in {\mathcal{V}}_i^{\mathrm{out}}\\ 0,& \text{others}\end{array}.$$

$$M{I}_i=\sum _{j\in {\mathcal{V}}_i^{\mathrm{out}}}{I}_{ij}-\sum _{s\in {\mathcal{V}}_i^{\mathrm{in}}}{I}_{si}.$$

$$c_{pq}(i)=⟨r_p(i)r_q(i)⟩,$$

$$\mathbf{C}=\mathbf{U}\Lambda {\mathbf{U}}^{\mathrm{T}},$$

$$f_{\mathbf{C}}(\lambda )=\frac{1}{M}\frac{\mathrm{d}n(\lambda )}{\mathrm{d}\lambda },$$

$$\mathbf{R}=\frac{1}{T}\mathbf{X}{\mathbf{X}}^{\mathrm{T}},$$

$$f_{\mathrm{RMT}}(\lambda )=\frac{Q}{2\pi }\frac{\sqrt{\left({\lambda }_{\max }^{\mathrm{RMT}}-\lambda \right)\left(\lambda -{\lambda }_{\min }^{\mathrm{RMT}}\right)}}{\lambda },$$

$${\lambda }_{\min }^{\mathrm{RMT}}=1+\frac{1}{Q}-2\sqrt{\frac{1}{Q}},$$

$${\lambda }_{\max }^{\mathrm{RMT}}=1+\frac{1}{Q}+2\sqrt{\frac{1}{Q}}.$$

Full text

††journal: Chaos, Solitons and Fractals

1 Introduction

Global food security, defined as the state where people are able to get sufficient, available, and nutritious food [1, 2], is always a worldwide concern [3, 1]. It is a challenge to support the demand of the growing population [4]. The latest State of Food Security and Nutrition in the World report, promulgated by five United Nations agencies, announces that food insecurity is escalating111See https://www.fao.org/3/cc0639en/cc0639en.pdf.. Climate change, conflict, and epidemics expose millions of people to hunger and malnutrition risks [5]. Extreme weather poses one of the most serious threats to food production [6], shifting regional consumption and trade patterns [5]. Conflicts like the current Russia-Ukraine crisis have the potential to exacerbate global food insecurity222Assessing Food Insecurity in 2022/23 at National and Sub-National Levels in 50 Countries Vulnerable to the Effects of the Ukraine-Russia Crisis (2022/08/07). See https://www.fao.org/3/cb9447en/cb9447en.pdf.. The epidemic of diseases also has a negative impact on the food system through labor losses and trade restrictions [7]. Under the shadow of the COVID-19 pandemic, global hunger worsened in 2020. Due to conflicts and the COVID-19 pandemic, export restrictions imposed by some economies, cause disruptions to staple food flows in the international market. Because these restrictions reduce external food supply, they have the potential to cause a spiral rise in global food prices [8], spreading supply instability to trading partners [9].

Food availability in an economy relies on domestic production, reserves, and trade. Food production reductions are compensated first by obtaining reserves and using food imports, and then by diminishing both domestic use and food exports [9]. It is now clearer that trade plays a crucial role in narrowing the gap between domestic consumption and supply. The international food trade provides access to approximately 24% of the food consumed by the global population [10]. If extreme events hinder food trade, the instability of the world market will exacerbate [11], thereby festering food shortages in resource-poor economies with food deficits [9]. Food trade within or between economies diversifies food supplies and rationalizes resource allocation, which assists in reducing the vulnerability of food systems to various types of shocks [7]. However, the formation of extensive and close trade relationships between economies may cause shocks to propagate to importers.

When an economy with a significant impact on the global food market restricts food exporting or importing as a result of an external disturbance, other economies and even the global food trade system are impacted. One then wonders how to identify critical economies and evaluate their influence on the trade system. To that end, we use complex networks as a tool to address this problem. We construct international crop trade networks (iCTNs) and investigate critical nodes in topology structures using international trade data sets from four staple crops. Our analysis is rooted in the core concept that the importance of a node is equal to its destructive effect on the network after it is deleted. The most common node importance metrics include degree scales, eigenvector scales, path scales, and so on. Different metrics depict various characteristics and demonstrate the influence rankings of nodes. Here, we summarize four types of indicators, including global metrics, local metrics, modular metrics, and metrics based on information theory. It helps us measure the influence of economies on the international crop trade networks in a holistic way. Despite the fact that numerous studies have used a specific metric based on a specific network property to identify key nodes, there is a critical knowledge gap regarding the correlation structure between different node importance metrics. Random matrix theory (RMT) is a successful approach to recognizing correlations among objects [12]. An open question is how to construct a comprehensive metric to measure the influence of an economy. We heuristically apply the random matrix theory for constructing optimal portfolios in the stock market to construct a composite node importance indicator.

It is widely assumed that measuring the trade influence of economies in the iCTNs is critical and advantageous in order to avoid massive trade blockages when shocks occur. In this paper, we apply various node importance metrics of four dimensions for each iCTN to identify critical economies in the food trade system. We specifically employ random matrix theory to examine the correlation structure of various node importance metrics. We find that the trade influence of economies in the iCTNs differs across node metric scales. However, some metrics that are based on the same direction of trade flows show special correlations. This new knowledge can inform a new mindset to obtain a composite node importance index. In this regard, we fictionalize the optimal eigenportfolio of eigenvalues to estimate an economy’s composite importance. Our approach does not distinguish between different types of shocks and simplifies the form of trade constraints. But our results highlight the importance of some economies in the global trade system and therefore point to the need to ensure unimpeded trade between critical economies.

The subsequent sections of this article are organized as follows: In Section 2, we provide an overview of the sources of our data and our research methods. Section 3 presents the findings of our analysis, which includes an examination of the correlation structure among different node importance metrics using random matrix theory, and the integration of 20 metrics into a single metric. In the final section, we summarize the key contributions of our paper, as well as its limitations and potential future directions.

2 Data and method

2.1 Data and iCTN construction

Most of the calories consumed by the world’s population come from four staple crops (maize, rice, soybean, and wheat) [10]. We collect bilateral food trade data from 1986 to 2020, provided by the FAOSTAT data sets on international trade flows from the Food and Agriculture Organization (FAO, http://www.fao.org). Note that the quantities and prices of crops vary across different crops, making it difficult to compare them on an equal footing. To solve this problem, we calculate caloric trade flows by multiplying calories per 1000 t and volumes.

We assemble the international crop trade networks, where economies are portrayed as nodes and trade relationships as links from importers to exporters. That is to say, we define the iCTN for a given crop $\mathrm{crop}$ as: $F^{crop}(t)=\left(\mathscr{V}^{crop}(t),W^{crop}(t)\right)$, where ${\mathscr{V}}^{crop}(t)$ is the set of all nodes (economies) and $W^{crop}(t)$ is the weight of links (i.e. the caloric trade flows). Having constructed the international maize, rice, soybean, and wheat trade networks from 1986 to 2020, we investigate and compare the characteristics and evolution of these different networks.

To track the dynamics of the iCTNs, we displaying the weights of links. In this paper, the total weight $W(t)$ is defined as the caloric values of imports and exports across all economies, that is $W(t)=s_{i}^{\mathrm{in}}(t)+s_{i}^{\mathrm{out}}(t)$. The network density $\rho$ depicts overall tightness. In the iCTNs, the closer the density is to 1, the more frequent the trade activities among economies are. Figure 1 shows the evolution of the weights and the density of four iCTNs from 1986 to 2020. Different colors stand for various types of crops. The up and down arrows indicate that the upward (downward) trend changes into the downward (upward) trend. Dotted lines with different colors depict three food-related crises. On the whole, the weights show an upward trend but experience small fluctuations, as presented in Fig. 1(a). We can observe that the dynamics of the weights vary across different crops. Notably, prior to 2018, wheat consistently held the top spot as the primary crop in the global food trade. In 2018-2020, maize’s weight overtook wheat’s. It is worth noting that the extent to which special events affect the weights is different in different iCTNs. The collapse of the Soviet Union in 1991 and the Asian financial crisis in 1996/1997 led to a decline in the trade volume of maize, soybeans, and rice, except for wheat. When looking at the evolution of network density, it is obvious that the links between economies were generally closer in the global food market from 1986 to 2020. Following the Southeast Asian financial crisis, the density of the international maize trade network has surpassed that of the international soybean trading network. Furthermore, the density increased significantly and ranked first following the 2008 food price crisis. The evolutionary properties of degree, strength, clustering coefficients, et al. have been investigated [13]. We have also studied the degree distributions and found that the in-degree and out-degree distributions have different forms in different iCTNs [14]. Since the import and export data of the international crop trade provided by FAO have been updated to 2020, we focus on the iCTNs in 2020, which are the latest trade networks.

Figure 2 is the Sankey diagram of the regional distribution of imports and exports of the four iCTNs in 2020. Figure 2(a-d) and (e-h) respectively depict major economies’ food export flows and import sources. Asia is the main food-importing region, while North America plays an important role in food exports. The regional patterns of food trade vary across crop sectors. The U.S. is an important exporter of the four crops, while Saudi Arabia has a significant impact on the import of maize and rice. Besides, China is a major maize and soybean importer.

Overall, the international food trade shows a booming trend with increasingly strong trade links. However, different crops have different trade patterns, and different economies have different impacts on the iCTNs. This motivates our focus on the trade influences of economies in different food trade networks in the subsequent study.

2.2 Economic importance metrics

2.2.1 Economic importance metrics based on the local structure of the iCTNs

There are many metrics to measure the importance of nodes in a network [15]. The larger the degree indicates that the node is more important in the network. However, the degree of a node in a large network is often too large to measure the node importance intuitively. Degree centrality is used as a direct measure. The greater the degree centrality of a node, the more neighbors it has in the network and the greater its influence [16, 17]. It is simply defined as

[TABLE]

Since the iCTNs we consider are directed networks, we use in-degree and out-degree centrality taking into account the directions of links [18]:

[TABLE]

and

[TABLE]

Degree centrality lays emphasis on the node itself. It considers that the more neighbors a node connects to, the greater the node’s role in risk propagation, but neglects the node’s neighborhood and the topology of the network. Semi-local centrality, on the other hand, considers that the influence of a node is also related to the influence of its neighbors [19]. It is defined as,

[TABLE]

where $\tau(i)$ denotes the set of nearest neighbor nodes of node $i$ and $k_{v}$ denotes the number of neighbors of node $v$. In directed networks, it is possible to construct in-semi-local and out-semi-local centrality,

[TABLE]

[TABLE]

where $\tau^{\mathrm{in}}(\cdot)$ and $\tau^{\mathrm{out}}(\cdot)$ respectively calculate the set of economies which food import from and which food export to for a given economy. $k_{v}^{\mathrm{in}}$ and $k_{v}^{\mathrm{out}}$ are respectively the in-degree and out-degree of node $v$.

The importance of a node depends on both the number and the importance of its neighboring nodes, so we use eigenvector centrality to measure the influence of a node, where the eigenvector is the eigenvector corresponding to the maximum eigenvalue of the network adjacency matrix [20]. It means,

[TABLE]

The eigenvector of maximum eigenvalues is given by solutions to the recursive equation,

[TABLE]

where $A=\{a_{ij}\}$ is the network adjacency matrix, $\mathbf{e}$ is the eigenvector corresponding to the largest eigenvalue, $e_{i}$ is the $i^{th}$ element in the vector $\mathbf{e}$, $\mathscr{V}_{i}$ is the set of neighboring nodes of node $i$, and $r$ is the number of iterations, thus we get the node $i$’s Eigenvector centrality. That is,

[TABLE]

Katz centrality is another metric for measuring network centrality and is commonly used in biological fields, for example, to assess the potential neural activity of neurons. It not only takes into account the centrality of the node itself, but also the centrality of its neighboring nodes [21]. Katz centrality measures the importance of a node $i$ by counting the number of neighboring nodes of the node $i$ and the number of all nodes connected to the node $i$ that pass through these neighboring nodes [22, 23],

[TABLE]

where $a_{ij}^{k}$ is the number of nodes passed from node $i$ to node $j$ along a path of length $k$, and $\alpha$ is the decay factor satisfying $0<1/\lambda_{\max}<1$ ($\lambda_{\max}$ is the maximum eigenvalue of the network adjacency matrix), which is set to 0.01 here. For node $j$, the farther it is from node $i$, the smaller its contribution to the Katz centrality of node $i$.

PageRank (PG) algorithm was proposed as an algorithm to calculate the importance of Internet pages, and then it was widely used to identify nodes in a directed network that have an important influence on the network structure. In the international food trade networks, the larger the value of a economy’s PageRank indicates that the more food import trade partners the economy has and the more food it exports to other economies. It suggests that the economy play a crucial role in the trade network. PageRank can be calculated by a simple iterative algorithm and corresponds to the principal eigenvector of the normalized network connectivity matrix [24].

Two metrics are introduces in the Hyperlink-Induced Topic Search (HITs) algorithm: Authorities (AU) and Hubs (HU). Similar to PageRank, the HITs algorithm was once used to rank web pages [25]. A node with a high authority value is pointed to by numerous other nodes with high hub values, while a node with a high hub value is pointed to numerous other nodes with high authority values. This approach has been utilized in a number of economic sectors as a tool for characterizing the network structure of web pages. The eigenvector for the matrix $A^{T}A$ in a network is the HITS authority vector, while the eigenvector for the matrix $AA^{T}$ is the HITS hub vector. Each economy in the iCTNs has an authority value and a hub value, which are related to the directed edges of the economy and other economies. Therefore, by calculating the Authority and Hub, a node’s importance can be determined.

The clustering coefficient (CC) is used to describe the aggregation of nodes in a network. The higher the value, the more important the node is in the network. A node’s clustering coefficient reflects the likelihood of connectivity with its neighbors. The clustering coefficient of a node in a directed weighted network can be determined in a number of different methods. In this paper, it is described as the ratio of the total number of directed triangles that node $i$ has actually formed to the total number of directed triangles that node $i$ could form [26], i.e.

[TABLE]

where $k_{i}$ is the total degree of node $i$ ($k_{i}=k_{i}^{\mathrm{in}}+k_{i}^{\mathrm{out}}$), node $s$ and $r$ are neighbors of node $i$, and $\hat{w}_{sr}=\frac{w_{sr}}{\max(w)}$, $\max(w)$ means the maximum of network weights.

2.2.2 Economic importance metrics based on the global structure of the iCTNs

The global features of the node are taken into account in the evaluation since the significance of a node in a network is tied to the network’s overall structure.

Betweenness centrality is the ratio of all shortest pathways between two nodes that travel through a specific node to all other shortest paths [27]. It gauges an economy’s capacity to function as an intermediary in the food trade networks, that is as a trade conduit between different countries or regions. An economy’s importance in the global food trade networks increases with its betweenness centrality. It is calculated as

[TABLE]

where $h_{pq}^{i}$ is the number of shortest paths from $p$ to $q$ through $i$, and $g_{pq}$ is the total number of all shortest paths from $p$ to $q$.

Closeness centrality is used to measure the proximity between a node and other nodes [28]. If a node is closer to other nodes, then the less it needs to rely on others when disseminating information. In the food trade networks, the greater the closeness centrality of an economy indicates that the economy is less likely to be constrained by other. We use the average shortest distance from node $i$ to all other nodes to represent the Closeness centrality. In a directed network, we can define in-closeness centrality (IC) and out-closeness centrality (OC) as

[TABLE]

[TABLE]

where $D_{i}^{\mathrm{in}}$ and $D_{i}^{\mathrm{out}}$ denote respectively the number of nodes that a node can reach $i$ and the number of nodes that a node $i$ can reach, and $d_{ji}$ and $d_{ij}$ denote the shortest path length from node $j$ to node $i$ or from node $i$ to node $j$, respectively.

The most influential nodes in a network with a community structure should have the following characteristics: serve as the community center and play a “bridging” role in each community. Therefore, the focus of the key node ranking problem should not be limited to the core nodes in the network, but also the nodes in the Structural Hole. Burt proposed the constraint index (CI) in 1992, which represents the ratio of the contribution of node $i$ to its neighbors to all its contributions [29]. In the food trade networks, the higher constraint coefficient of a node, the greater the bridging role of the node in the community structure.

2.2.3 Economic importance metrics based on the modular structure of the iCTN

Community structure appears to be common in a social network [30]. The gathering nodes in a community show that there is a higher density of links between them than other nodes outside the community [31]. Modularity is an important feature of a network and a part of agglomeration algorithms for detecting community structure. It is plausible that nodes with the same role should perform analogous topological characters [32]. Thus we can extract the significant information about the role of nodes by considering properties of nodes within module. More often than not not, the role of a node in a network can be measured by its within-module degree (IM) and participation coefficient (PC) [32].

Within-module degree (IM) measures the extent of connection of a node to other nodes in their module [32], which is suitable for portraying the relationship between economies and gauging an economy’s influence within the module. For a given network, within-module degree is:

[TABLE]

where $\kappa_{i,m_{i}}$ is the number of links of node $i$ to other nodes in its module $m_{i}$, $\overline{\kappa}_{m_{i}}$ is the average of $\kappa_{i}$ over all nodes in $m_{i}$, and $\sigma_{\kappa_{m_{i}}}$ is the standard deviation of $\kappa_{i}$ in $m_{i}$. The value of $IM_{i}$ increases with the number of trade links of the economy $i$ to other economies in its module $m_{i}$. It indicates that the economy plays a pivotal role in the modular structure.

Occasionally, nodes with the same $IM_{i}$ may arise different roles. Participation coefficient of node $i$ is used to measure the uniformly distributed of its links among all the modules [32],

[TABLE]

where $N_{M}$ is the number of modules involving node $i$.

To measure how well-influence of nodes outside the module to node $i$, we apply outside-module degree (OM) [33],

[TABLE]

similarly, where $\kappa_{i,m_{i}}^{{}^{\prime}}$ is the number of links of node $i$ to other economies outside its module $m_{i}$, $\overline{\kappa}_{m_{i}}$ is the average of $\kappa_{i}$ over all nodes outside $m_{i}$.

2.2.4 Economic importance metrics based on mutual information metric

The importance of a node depends on not only the network topological structure, but also the strength of the connection between nodes. Therefore in this paper we consider mutual information metric that is an essential point in Information theory to access the ability of nodes to transit information. Shannon argues that information is something useful obtained by message receiver through passing messages , thus eliminating the “uncertainty” that existed before the communication [34]. It is widely believed that the iCTNs are complex communication systems. We can measure the importance of nodes based on information theory. It is plausible to surmise that the measure of a node’s importance in a network can be fulfilled according to the amount of information contained in the node [35].

For a undirected and unweighted network, mutual information of a node depends on node degree,

[TABLE]

Information of node $i$ is the sum of mutual information between it and other node, that is,

[TABLE]

Since the iCTNS are directed and weighted, the directions and weights of links are crucial. Based on the above method, we consider node information of a directed and weighted network. A node may be both a transmitting and a receiving node at the same time. Therefore, we define the probability of out-links and in-links of node $i$. Since $w_{ij}$ is the weight of the link from node $i$ to $j$, the probability of out-links is the ratio of $w_{ij}$ to all weighted links transiting information from node $i$ that is equal to out-strength of node $i$. We calculate the probability of out-links as follows,

[TABLE]

Similarly, we obtain the probability of in-links as follows,

[TABLE]

Then the mutual information $I_{ij}^{w}$ from node $i$ to $j$ is,

[TABLE]

So, the information embraced in node $i$ is the sum of the mutual information from node $i$ to all $i$-pointing nodes minus the sum of the mutual information from all nodes pointing to $i$,

[TABLE]

It goes without saying that nodes possessing more information are more important in a network. In the iCTNs, mutual information is transmitted between economies $i$ and $j$ if there is a link between them. If there is no link, their mutual information is 0. As a result, the quantity of reciprocal information both reflects the connection and importance of economies in the trade network.

2.3 Correlation structure of node importance metrics

Random matrix theory is often used to study financial markets [36]. Here, we use it to analyze the correlation structure between different node importance metrics. Based on the above 20 node importance metrics, we obtained the rankings of trade influence of selected economies in four iCTNs from 1986-2020. We set the rankings of an economy $i$ for a given iCTN in time $t$ as $NI_{i}^{crop}(t)$ and analyze the correlation structure of different ranking based on each importance metric. Our approach works for all iCTNs in any time $t$, so time $t$ and crop type $\mathrm{C}$ are ignored to predigest the formula.

We obtain a matrix of node importance rankings for each year of the four iCTNs from 1986-2020 and calculate the correlation coefficients of importance rankings among different nodes,

[TABLE]

where $r_{p}(i)$ is the importance ranking of $i$-th economy based on the $p$-th node importance index.

By resolving the following equation, we procure the eigenvalues and related eigenvectors of the correlation coefficient matrix $\mathbf{C}$ for various grains across various years,

[TABLE]

where $\mathbf{U}$ represents matrix of the eigenvectors $u_{i}$, $\Lambda$ represents the diagonal matrix of the eigenvalues $\lambda_{i}$, $\mathbf{U}^{\mathrm{T}}$ is the transpose of $\mathbf{U}$. The probability density function $\Lambda$ is

[TABLE]

where $n(\lambda)$ means the number of eigenvalues which are smaller than $\Lambda$ of the matrix $C$. Thus, we construe a random matrix as,

[TABLE]

where ${\mathbf{X}}$ is an $M\times N$ matrix that incorporates $M$ nodes and $N$ random elements with zero mean and unit variance. Here $N=20$, and $M$ is the number of nodes in the network each year. When $M\rightarrow\infty$ and $N\rightarrow\infty$, if $Q=N/M>1$, the probability density function $f_{\mathrm{RMT}}(\lambda)$ of $\lambda$ of $R$ is given by [37, 38, 39],

[TABLE]

where $\lambda\in\left[\lambda_{\min}^{\mathrm{RMT}},\lambda_{\max}^{\mathrm{RMT}}\right]$, and $\lambda_{\min}^{\mathrm{RMT}}$ and $\lambda_{\max}^{\mathrm{RMT}}$ are respectively the minimum and maximum eigenvalues of $\mathbf{R}$, which can be calculated as follows,

[TABLE]

and

[TABLE]

2.4 Comprehensive node importance metric

The inverse participation ratio (IPR) is used to describe the importance of elements of the matrix [40, 41], which measures the contribution of the $l$th element of $u^{k}$ to the eigenvectors. Likewise, IPR has also been used to measure the degree to which the eigenvector deviates from the RMT results. The deviation of the eigenvector is inversely correlated with IPR, meaning that the lower the IPR, the higher the eigenvector vector deviation. The inverse participation ratio $I^{k}$ is given by,

[TABLE]

where $\left\{u_{l}^{k}:l,k=1,\cdots,M\right\}$ denote the $l$-th components of the $k$-th eigenvector $u^{k}$. If all the eigenvector elements in $u^{k}$ are equal and satisfy $u^{k}_{l}=1/\sqrt{M}$, then $I^{k}=1/M$. When one of the eigenvector elements of $u^{k}$ fulfills $u^{k}_{l}=1$ and all other eigenvector elements of $u^{k}$ are equal to 0, then $I^{k}=1$. As a result, the inverse participation ratio (IPR) can determine whether the number of eigenvector elements’ inverse is noticeably non-zero.

In the financial markets, eigenportfolios are frequently built for each eigenvalue to reflect changes in the returns of a composite portfolio [42]. On this basis, we develop the composite index for assessing an economy’s importance in global food trade networks. By constructing the corresponding eigenportfolio, we can obtain the composite importance index for an economy $i$ given the eigenvalue $lambda_{k}$,

[TABLE]

where $\sum_{j=1}^{M}u_{j}^{k}r_{j}(i)$ is the projection of the economy’s importance ranking series $r_{j}(t)$ on the $k$-th eigenvector $u_{j}^{k}$. The optimal eigenportfolio is found by scrutinizing the relationship between economies’ importance rankings based on different eigenportfolios and average rankings $\left\langle r_{j}(i)\right\rangle$. Finally, we equate an economy’s composite importance ranking with the results of the optimal eigenportfolio.

3 Empirical results

3.1 Ranking of economic influence in the iCTN

Node importance metrics are widely used to identify critical economies in different trade networks [43, 44, 45]. Previous published studies are limited to one indicator or one type of indicator but rarely compare the various types of indicators. Meanwhile, fewer studies on food trade networks apply different indicators and comparatively analyze the trade influence of economies in different kinds of food networks. We calculate the influence of economies in different iCTNs from 1986 to 2020 based on the nodal importance indicators presented in Section 2. The trade impact of economies tends to change over time and varies across crops. We focus on economies’ importance in maize, rice, soybean and wheat trade networks and show ranking results in Table 1.

Comparing the ranking results of different indicators, the four iCTNs have some interesting similarities. First, the indicators about degree scales based on the local structure of networks (in and out-degree centrality, in and out-semi-local centrality) only consider the number of export and import partners of the economies and do not take into account the weights and clustering distribution of trade links. As a result, the ranking results obtained based on the strength indicators are different from those obtained based on the degree indicators. However, the ranking results based on in-degree centrality and in-semi-local centrality are similar, as are those based on out-degree centrality and out-semi-local centrality. Secondly, we can see that the rankings obtained by in-degree centrality, in intensity, eigenvector centrality, in-closeness centrality, PageRank, authorities, and in semi-local centrality as metrics are relatively comparable. Due to the fact that they focus primarily on the economy’s food imports, these indicators have a significant correlation. Similar features exist for the metrics that mainly consider the economy’s export profile. The ranking results based on out-degree centrality, out-strength, Hub, out-closeness-centrality, and out-semi-local centrality are more similar. Likewise, rankings based on metrics that consider the distribution of other nodes connected to the nodes in the network have special properties, such as clustering coefficient, Katz centrality, within-module degree, participation coefficient, and out-module degree. The ranking results computed by these metrics vary widely. However, the ranking results of metrics based on within-module degree and outside-module degree are very similar.

The trade influence of economies differs in different iCTNs. The trade influence of economies differs across different iCTNs. To get a clearer picture of food trade in key economies, we select three key economies in each iCTN in 2020 and portray the evolution of the in and out-degrees as well as the in and strengths of these economies from 1986-2020. Both the U.S.and the Netherlands have a significant impact on the maize and soybean trade, which can be seen in Panel A and Panel C of Table 1, present that the United States exports large amounts of maize and soybean to many economies. The Netherlands has a significant impact on the international maize trade network mainly because of its substantial maize imports from numerous economies. Interestingly, the Netherlands has a larger in-degree in the soybean trade network, i.e., it has many import trading partners, but its import volume is not large compared to China. For rice, Italy and India occupy a prominent position. The possible reason for this case is that India is the main rice exporter, and a large number of economies are highly dependent on rice imports from India. Italy imports and exports rice, with imports outgrowing exports (Fig. 3(b)). China is one of the world’s largest soybean markets, with many trade partners. Given the fact that China imports a lot of soybeans from the United States, China plays an important role in the global soybean import trade. British has a significant influence on the international wheat trade network due to its wheat imports. Concerning wheat exports, we found that the United States and Russia have an important impact on the wheat network. The possible explanation could be that these economies are principal wheat planting areas and suppliers. Based on different ranking comparisons, economies play divergent roles in different iCTNs. The trade influence of an economy is associated with its crop production. Trade could compensate for the gaps between domestic crop production and consumption of an economy. Thus an economy’s demand for external food supplies determines its influence in the iCTNs. It is necessary to discuss the contribution of a particular economy to global food trade and food security across different crop types. Furthermore, the ranking results also show that some economies are indeed important, for instance, the United States and the Netherlands are identified as key economies in several food trade networks. The European economies have a relatively large influence on the global food trade. Risks to the global food supply must be tracked, and this requires keeping a watch on food production and trade in these economies.

3.2 Correlation structure among node importance rank metrics

Figure 4 shows the cross-correlation structure among 20 node importance rank metrics of the iCTNs. In each Heat map, blocks of related hues are easy to spot. We find that there are generally two significant clusters of indicators. One cluster mainly contains node metrics related to import quantities, including in-closeness centrality, in-semi-local centrality, and eigenvector centrality. The other cluster mainly contains node metrics related to export quantities, including out-closeness centrality, out-semi-local centrality, betweenness centrality, and Hubs. The correlation between indicators in the same cluster is high.

Figure 5 shows the distribution of the correlation coefficient $c_{ij}$. The histogram is the distribution of the probability density $f(c_{ij})$ of $c_{ij}$, and the curve is the corresponding kernel density distribution. As can be seen from Fig. 5, positive correlations are rifer in the correlation structure of these indicators. Further, the kernel density distribution indicates that there are two peaks in the correlation structure for the maize, rice, and soybean trade networks. It is clear that some of the indicators do in fact significantly correlate with one another.

To further characterize the correlation structure of node important metrics, we adopt hierarchical clustering based on the leaf ordering algorithm to search for clusters. Figure 6(a-d) present cluster identification of rankings of node importance in 2020. We find that these metrics can be roughly divided into three groups. One is determined by the import scale, such as in-degree centrality, in-strength, in-closeness centrality, in-semi-local centrality, eigenvector centrality, Authorities, PageRank, within-module degree, outside-module degree, and participation coefficient. One is based on the export scale, namely, out-degree centrality, out-strength, out-closeness centrality, out-semi-local centrality, betweenness centrality, and Hubs. The remaining less relevant indicators are grouped into one group, i.e., those that present different characteristics in the iCTNs, such as constraint coefficient, clustering coefficient, Katz centrality, and mutual information. These clusters are not special cases in 2020. As shown in Fig. 6(e-h), out-degree centrality, out-strength, out-closeness centrality, out-semi-local centrality, betweenness centrality, and Hubs (indicators marked under the orange line) are always in the same cluster, except for the occasional absence of betweenness centrality and Hubs. Indicators based on import scales are typically classified into the same cluster. Other indicators’ behaviors are impenetrable. The results of clustering are also aligned with the results of our correlation analysis for these indicators.

3.3 Distribution of eigenvalues

Different nodal importance indicators measure the influence of an economy in the iCTNs from different perspectives. It is comparatively biased to assess the importance of an economy using a single indicator. According to the correlation analysis above, it is worth mentioning that some indicators correlate with indicators based on the same flow directions. To probe into the correlation structure of these node importance metrics, we imitate the random matrix theory for constructing optimal eigenportfolios in the stock market to reduce the dimension and construct a composite node importance indicator.

We obtain eigenvalues and their corresponding eigenvectors of correlation matrices of economies’ trade influence rankings in the four iCTNs. The largest eigenvalues $\lambda_{\max}^{\mathrm{RMT}}$ of four correlation matrices in 2020 are respectively 1.7705, 1.8506, 1.8321 and 1.7983. The smallest eigenvalues $\lambda_{\min}^{\mathrm{RMT}}$ are respectively 0.4481, 0.4091, 0.4179, 0.4343. We calculate 20 eigenvalues of the empirical correlation matrix $\mathrm{C}$ and sort $\lambda_{i}$ in descending order ($\lambda_{i}>\lambda_{i+1}$). For the international maize trade network, the smallest eigenvalue $\lambda_{20}=0.0014$, which is about 1/320 of $\lambda_{\min}^{\mathrm{RMT}}=0.4481$, and the largest eigenvalue $\lambda_{20}=10.6541$, which is about 6 times larger than $\lambda_{\max}^{\mathrm{RMT}}=1.7705$. Similar cases about the relationships between empirical correlation matrices and random matrices exist in the other three iCTNs. Figure 7 compares the empirical distribution $P(\lambda)$ of these eigenvalues of $\mathrm{C}$ with the theoretical distribution $f_{\mathrm{RMT}}(\lambda)$ based on the random matrix $\mathrm{RMT}$. We find that most of the eigenvalues of $\mathrm{C}$ deviate from $f_{\mathrm{RMT}}(\lambda)$. For maize, rice, soybean and wheat, 3, 3, 3 and 3 minimum eigenvalues are respectively smaller than $\lambda_{\min}^{\mathrm{RMT}}$, 13, 12, 12 and 12 maximum eigenvalues are respectively larger than $\lambda_{\max}^{\mathrm{RMT}}$, and 4, 5, 6 and 5 eigenvalues are covered within $f_{\mathrm{RMT}}(\lambda)$.

To capture the deviation of the correlation matrix $\mathrm{C}$ from the results of the random matrix $\mathrm{RMT}$, we present the inverse participation ratio (IPR). Figure 8 shows the inverse participation ratios of all eigenvalues of the four correlation matrices. The average inverse participation ratios are 0.194, 0.222, 0.214, and 0.222, respectively, indicating that the average contribution is 5. The inverse participation ratios $I^{1}$ corresponding to the largest eigenvalue $\lambda_{1}$ are 0.065, 0.060, 0.062, and 0.062, respectively, and their inverse is about 16, indicating that about 16 indicators contribute to the eigenvector $u_{1}$. The inverse participation ratios corresponding to the second largest eigenvalue, $\lambda_{2}$, which falls in the upper $\lambda_{\max}^{\mathrm{RMT}}$, are 0.734, 0.066, 0.074, and 0.069, indicating that about 14 indicators contribute to the eigenvector $u_{2}$. The inverse participation ratios corresponding to the other eigenvalues vary and generally show that the smaller the eigenvalue, the larger the corresponding inverse participation ratio. This indicates that the major eigenvalues dedicate most information to relative eigenvectors.

We present all the eigenvectors in B. There are information contents in the components of eigenvector $u_{1}$ (associated with the largest eigenvalue $\lambda_{1}$), eigenvector $u_{2}$ (associated with the second largest eigenvalue $\lambda_{2}$), eigenvector $u_{3}$ (associated with the third largest eigenvalue $\lambda_{3}$), and eigenvector $u_{20}$ (associated with the smallest eigenvalue $\lambda_{20}$), as shown in Fig. 9. Specifically, Fig. 9(a-d) provide a detailed breakdown of the eigenvectors $u^{1}$ corresponding to eigenvalues for the maize, rice, soybean, and wheat networks. One notable finding is that the eigenvector of the largest eigenvalue contains the majority of the information, and most of the components of the eigenvectors have the same signs. Intriguingly, the components of $u_{2}$ (shown in Fig. 9(e-h)) and $u_{3}$ (shown in Fig. 9(i-l)) corresponding to the same group of indicators largely have the same signs. Additionally, the components of $u_{2}$ associated with indicators based on the import and export scale have opposite signs. Furthermore, the components corresponding to the indicators $KC$ and $CC$ (i.e., the sixth and seventh components) show the opposite sign with respect to $u_{1}$ for maize and wheat (as seen in Fig. 9(a) and (d)). These observations suggest that the distribution of components corresponding to the eigenvectors of the correlation matrix $\mathrm{C}$ is likely related to the performance of the corresponding indicator that assesses the importance of economies. For example, Fig. 9(e,g,h) clearly demonstrate that the absolute values of the eigenvector components corresponding to $EC$ and $IL$ are large, while others are close to zero, indicating a significant correlation. Moreover, the analysis reveals that $u_{20}$ covers less information, and components corresponding to highly relevant metrics tend to exhibit similar features. Overall, these findings shed new light on the relationship between eigenvector components and economic indicators. They provide a compelling illustration of the extent to which the eigenvector components can reveal important insights into the behavior and performance of different node importance metrics, thereby enhancing our understanding of the economies’ impact on the iCTNs.

3.4 Composite trade influence of economies

The results obtained from analyzing the correlation coefficient matrix $\mathrm{C}$ and $\mathrm{RMT}$ demonstrate that the information provided by the indicators is predominantly contained in the largest and second largest eigenvalues, and this finding holds not only for the 2020 data but also for data from 1986-2020. Specifically, our analysis reveals that the first and second largest eigenvalues of the correlation matrix consistently exceed the maximum eigenvalue of the random matrix, while the third to twentieth eigenvalues are consistently smaller, except for some cases where the third largest eigenvalue slightly exceeds the maximum eigenvalue (Fig. B1). Therefore, we employ the method in line with constructing eigenportfolios based on the random matrix theory in financial markets. We calculate the largest eigenvalue combination $\mathrm{G}^{1}$, and the second largest eigenportfolio $\mathrm{G}^{2}$ and compare them with the average node importance rankings $\left\langle{NI}\right\rangle$ to obtain the optimal eigenportfolio as the composite node importance metric $\mathrm{CNI}$. In recognition of the existence of indicator clusters, we divide the eigenvalues and corresponding eigenvectors into three categories, leading to the construction of three types of eigenportfolios $\mathrm{G}^{\mathrm{in}}_{k}$, $\mathrm{G}^{\mathrm{out}}_{k}$, and $\mathrm{G}^{\mathrm{third}}_{k}$, which correspond to the three categories of node importance metrics identified in Section 3.2.

Figure 10 illustrates the relationship between the largest eigenvalue portfolios and the average node importance rankings, while Fig. 11 displays the relationship between the second-largest eigenvalue portfolios and the average node importance rankings. A notable observation arising from Fig. 10 is that both $\mathrm{G}^{\mathrm{in}}_{1}$ and $\mathrm{G}^{\mathrm{out}}_{1}$ exhibit a linear relationship with $\left\langle{NI}\right\rangle$, with the exception of $\mathrm{G}^{\mathrm{third}}_{1}$, particularly for maize (Fig. 10(a)) and wheat (Fig. 10(d)). The results presented in Fig. 11 are comparable to the above descriptions, except that $\mathrm{G}^{\mathrm{in}}_{1}$, $\mathrm{G}^{\mathrm{out}}_{1}$, and $\mathrm{G}^{\mathrm{third}}_{1}$ are more proximate to $\left\langle{NI}\right\rangle$. These findings suggest that the largest eigenportfolio embodies a collective effect of multidimensional node importance metrics. Taken as a whole, these results encourage us to convert the maximum eigenportfolios into composite node importance metrics, which enable us to rank economies’ composite trade influence based on three types of composite node importance metrics.

Table 2 presents the top ten most important economies in the four iCTNs for 2020, using composite node importance metrics. It is evident that the ranking results based on $\mathrm{CNI}^{\mathrm{third}}$ are perplexing and divergent. Hence, we focus on the ranking results based on $\mathrm{CNI}^{\mathrm{in}}$ and $\mathrm{CNI}^{\mathrm{out}}_{1}$. Notably, European economies have a significant influence on the iCTNs, with the Netherlands, Britain, and Germany being particularly influential.

For a more detailed analysis of the factors contributing to the influence of economies, we have plotted the regional distribution, crop production ratio (the ratio between the crop production of a given economy and that of all economies), population ratio (the ratio between the population of a given economy and that of all economies), and GDP per capita of the key economies in Fig. 12. According to the ranking results based on $\mathrm{CNI}^{\mathrm{in}}$, economies with low crop production exert a greater impact on import trade, as exemplified by the Netherlands, the UK, and Spain importing maize (Fig. 12(a)), and Germany, Netherlands, and Italy importing rice (Fig. 12(b)). It should be noted that all of these are developed economies with high GDP per capita. At the same time, Asian economies also wield significant influence on food imports. This could be attributed to their low food production, such as Japan importing maize, the United Arab Emirates importing maize, rice, and wheat, and Israel importing wheat. Alternatively, it could be due to the large population size leading to gaps between local production and domestic food demand, as is the case with China’s importation of soybeans (Fig. 12(c)).

According to ranking results based on $\mathrm{CNI}^{\mathrm{out}}$, we can discern that economies with significant export influences are typically those with higher food production or smaller populations. For instance, the United States and Brazil, with their high maize (Fig. 12(e)) and soybean production (Fig. 12(g)), India, with its high rice (Fig. 12(f)) and wheat production (Fig. 12(h)), and Russia, with its high wheat production (Fig. 12(h)), have significant influence on the export trade. Some European economies with low crop production may export more crops, owing to their small population and food surpluses. Notably, France and the Netherlands, without much maize production (Fig. 12(e)), Italy and Spain, without much rice production (Fig. 12(f)), Austria and Germany, without much soybean production (Fig. 12(g)), and Lithuania, without much wheat production (Fig. 12(h)), play important roles in the trade. The result is somewhat counterintuitive that some economies without crop production are still identified as key economies based on $\mathrm{CNI}^{\mathrm{out}}$. For example, the United Kingdom without rice and soybean production in 2020 (Fig. 12(f, g)), and the Netherlands without soybean production in 2020 (Fig. 12(g)), remain vital to the trade network. These two economies have higher values of $OC$, which indicates their significant impact on the structure and function of the iCTNs. European economies, in general, occupy prominent trade positions in the iCTNs, and Asian and North American economies also hold important roles. Furthermore, economies with high food production wield significant influence over export trade, whereas those with insufficient food production have considerable sway over import trade.

4 Discussion

Prior studies have noted that food security is a grave concern in the wake of extreme weather, conflict, and epidemics. External food supply through international trade is of fundamental importance for economies with insufficient domestic production. Trade restrictions imposed by economies with a significant impact on food trade may exacerbate the volatility in global food markets. It has rekindled political and scientific interest in measuring economies’ trade influence. Complex network analysis has been proposed to investigate food trade patterns. In this paper, we assemble international crop trade networks using maize, rice, soybean, and wheat trade datasets from 1986 to 2020. Our initial objective is to identify key economies that have a substantial impact on global food trade. We evaluate the importance of economies in the iCTNs and identify significant economies using multidimensional node importance metrics. Through analyzing the correlation structure of different node importance metrics based on random matrix theory, we incorporate 20 metrics into a single composite metric. Our analysis yields some intriguing results.

Firstly, economies’ impact on the iCTNs depends on the focus of node important metrics. The findings unequivocally indicate that how important economies are in the iCTNs is a multidimensional problem. It motivates our focus on the correlation structure of those indicators. Despite indicators in our work gauging economies’ importance from different aspects, some of them have significant correlations. Based on correlation analysis and cluster identification, we classify indicators into three groups. One is based on the import scale such as in-degree centrality, in-strength, in-closeness centrality, in-semi-local centrality, eigenvector centrality, Authorities, PageRank,within-module degree, outside-module degree, and participation coefficient. Another is based on the export scale, namely, out-degree centrality, out-strength, out-closeness centrality, out-semi-local centrality, betweenness centrality, and Hubs. The remaining less relevant indicators are grouped into one category, such as constraint coefficient, clustering coefficient, Katz centrality, and mutual information. These findings facilitate our understanding of node importance metrics and enlighten us to construct composite metrics.

In terms of composite node importance metrics following optimal eigenportfolios in the financial market, we present the ranking characteristics of influential economies in the iCTNs. It should be pointed out that European economies have a significant influence in the iCTNs, especially the Netherlands, Britain and Germany. In addition, economies with low crop production have a greater influence on the import trade. At the same time, Asian economies also have an important influence on food imports. A possible explanation for this might be the low food production. Another possible explanation is that the large population results in gaps between local production and domestic food demands. Another important finding is that influential economies on food exporting are generally those with higher food production or smaller populations. What is counterintuitive is that some economies without crop production are still identified as key economies since their trading partners are important in the network.

Finally, in view of the unique trade patterns of different crops, economies’ trade impact differs across crops. These findings might be explained by the fact that economies’ food trade is tied to crop diversity and nutrition demand. Therefore, it is necessary to talk about the contribution of an economy to global food trade and food security across different crop types.

Since the criteria for importance are diverse, identifying key nodes is not a trivial job. It is impossible to find a perfect index to quantify a node’s importance. Our motivation for this paper is to make extensive empirical comparisons with multi-dimensional metrics. Thus, we can go a step closer to a comprehensive grasp of the similarities and differences across various approaches. Despite we just qualitatively evaluate node importance metrics instead of designing a convincing index, our study nevertheless helps identify vital nodes in every situation. Further investigation about objective functions considering metrics’ performances should, however, be the subject of future work.

This paper reaffirms the existing research measuring node importance based on different methods. Overall, vital node identification is challenging and meaningful. We find some approaches have many similarities and dissimilarities using the network theory and random matrix theory. We also find that European economies have a significant impact on the iCTNs. Additionally, economies with poor crop production play a major role in the import trade, whereas economies with higher food production or smaller populations are crucial to the export trade.

CRediT authorship contribution statement

Funding acquisition, Wei-Xing Zhou; Investigation, Yin-Ting Zhang; Methodology, Yin-Ting Zhang and Wei-Xing Zhou; Supervision, Wei-Xing Zhou; Writing – original draft, Yin-Ting Zhang and Wei-Xing Zhou; Writing – review & editing, Yin-Ting Zhang and Wei-Xing Zhou.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data Availability

Publicly available datasets were analyzed in this study. This data can be found here: https://www.fao.org.

Acknowledgements

This work was partly supported by the National Natural Science Foundation of China (72171083), the Shanghai Outstanding Academic Leaders Plan, and the Fundamental Research Funds for the Central Universities.

Appendix A Correlation of the influence indicators

Appendix B Eigenvalues and components of the corresponding eigenvector