The Review of Ecological Network Indicators in Graph Theory Context: 2014–2021

Graph theory (GT) is extensively applied in the ecological network analysis. This review study aimed to examine GT in the field of ecological network analysis based on the following questions: In what areas are the articles focused?, what indexes or graph-based indicators have been thus far utilized in ecological network analysis?, and what aspects of ecological network analysis have been less considered in terms of the use of the GT indicators? To address these questions, a systematic literature review was conducted and the results showed that most of the articles in this field had been fulfilled in China, the United States, and France. This theory could have implications for more research on plants and mammals. In addition, 118 indicators were identified in the field of GT in the ecological network analysis. Among these indicators, the probability of connectivity (PC) and an integral index of connectivity (IIC) had been consistently exploited in most articles. Moreover, the results revealed the increasing trend of introducing the new indicators of GT to ecological network analysis, suggesting the applicability of GT in this context. Despite the importance of ecological network resilience, it has been less reflected from the GT perspective while it can be useful and efficient in analyzing the sustainability of ecological networks within this framework. The current trend of exploiting the GT indicators delineates three future lines of development, viz. (1) the GT use more widely in ecological network analysis, (2) emerging new and more precise indexes, and (3) new concerns mainly examining ecological network resilience. China (16.67%), the USA (13.15%), and France (13.15%) have the highest number of studies in term of investigation the geographical distribution. Total of 118 indexes were used 485 times and the probability of connectivity, integral index of connectivity and least-cost-path algorithm is widely used in graph theory context. The indexes PC (probability of connectivity), IIC (integral index of connectivity), and LCP (least-cost-path algorithm) extensively used in different studies. The ecological networks resilience in graph theory milieu is the major identified gap. China (16.67%), the USA (13.15%), and France (13.15%) have the highest number of studies in term of investigation the geographical distribution. Total of 118 indexes were used 485 times and the probability of connectivity, integral index of connectivity and least-cost-path algorithm is widely used in graph theory context. The indexes PC (probability of connectivity), IIC (integral index of connectivity), and LCP (least-cost-path algorithm) extensively used in different studies. The ecological networks resilience in graph theory milieu is the major identified gap.


Introduction
Over the past 20 years, graph theory (GT) has been one of the most widely used tools that ecologists have used to study ecological networks (Bishop-Taylor et al. 2018;Ruiz et al. 2014;Tulbure et al. 2014). This theory is widely used, due to its high application in providing a functional view of networks and also the lack of need for a lot of data (Calabrese and Fagan 2004;Foltête 2018;Galpern et al. 2011). Indeed, GT provides a framework for visualizing and analyzing ecological connections at different spatial-temporal scales (Grech et al. 2018;Rayfield et al. 2011).
An ecological network is constituted based on a systemoriented approach, which explains the systematic interaction of an ecosystem by considering its structure and function. Hereupon, it illuminates the overall ecological constitutions and their invisible relationships (Borrett and Scharler 2019; Fath et al. 2019Fath et al. , 2007. As one part of ecological networks (Borrett et al. 2014), GT consists of two key elements, viz. nodes (representing discrete habitats) and links (denoting functional connections or environmental flows between the nodes) (Minor and Urban 2008;Urban et al. 2009). It also applies to simplify complex ecosystems by creating their mathematical representations in the form of easy to understand vertices, edges, and flows in graph structures (Gross and Yellen 2005).
Using the methods and tools provided by GT, researchers and especially ecologists can study, understand, and analyze complex environments more effectively and more conveniently. Therefore, it provides appropriate grounds for protecting, improving, or creating ecological networks (Foltête 2018;Foltête et al. 2014), as well as managing and reducing the unfavorable impacts of development on nature (Clauzel et al. 2015b;Foltête et al. 2014).
Over the years, GT has been developed and used to analyze many ecological networks of the plant (Huang et al. 2016;Tambosi et al. 2014), animal species (Meurant et al. 2018;Sahraoui et al. 2017) and context (Masselink et al. 2017) in both terrestrial (Mikoláš et al. 2017) and aquatic (Stewart-Koster et al. 2015) ecosystems (Grech et al. 2018;Urban et al. 2009). This study focuses mainly on the habitat network because it focuses more on the ecological connectivity network.
Ecological networks are also influenced by landscape patterns. Spatial-temporal changes in landscape connectivity also determine the characteristics of ecological networks and their functions (Huang et al. 2021a, b). As a result, landscape connectivity analysis is often assumed as an effective measurement of network changes in a landscape scale, especially in habitat studies. Optimization, improvement, and protection of an ecological network also rely on landscape connectivity. In other words, the diminution of landscape fragmentation plays a critical role in enhancing ecological network function and integrity. Accordingly, various indicators have been thus far developed to assess landscape connectivity and fragmentation in the context of GT (Huang et al. 2021a, b;Liu et al. 2014;Saura et al. 2011).
There have been several extensive reviews of studies in the field of ecological network analysis. These include the works of Janssen et al. (2006), which provided a framework for the study of socio-environmental systems with a focus on interactions between the components of these systems, Saint-Béat et al. (2015), which reviewed the theories that link food web structure and performance to the sustainability of ecosystems. Borrett et al. (2018) provides a systematic review of studies in the field of ecological network analysis from 2010 to 2016, and Kleinman et al. (2019), which identified and described the ecological consequences of discrete forest disturbance events involved in compound interactions (Kleinman et al. 2019).
While GT has been used in various areas of ecological analysis for many years (Ernst 2014b;Hejkal et al. 2017;Hofman et al. 2018;Loro et al. 2015;Zhang et al. 2019), few studies have been conducted on its application in ecological networks. In this regard, we can refer to the research conducted by Borrett and his colleagues in 2014 in the field of Network ecologists. Which has documented the emergence of network ecology and identified the diversity of topics discussed in this field, as well as mapping the scientific cooperation between scientists specializing in this field. They have used three terms to identify research, one of which is GT (Borrett et al. 2014). In that study, they examined GT in a broader format because the study aimed was to identify and determine the scope of the network environment.
Considering the extensive use of GT and graph-based indexes in the field of ecology, the various aspects of this use need to be explored more systematically, and special focus is needed on the concept of GT in network analysis in the research. Therefore, this study aimed to investigate the use of graph-based indexes in the studies of ecological networks and also aspects of the studies of ecological networks for which the indicators of GT have been presented or less considered. For this purpose, given that in recent years the use of GT in the ecological network has been growing, with considering that the results and methods that have been presented in this field recently are updated to be. The authors reviewed the indicators used in such studies, which were published between 2014 and 2021. For achievement to the main aim, the following research questions are specifically addressed: (1) What is the geographical distribution of studies and their relationship with each other? (2) In the field of ecological networks, what indicators of GT have been used and what are the most widely used indicators? (3) What aspects of ecological network studies have been less considered in the usage of GT indicators? This was done by conducting a systematic review of relevant articles published in this period and specifically the studies that have used GT.
The searched databases were Google Scholar, Scopus, and ISI Web of Knowledge. Because we try to search the ecological network in GT context. This search was narrowed to articles written in English in which GT has been used in the study of ecological networks.
To find as many potentially eligible articles as possible, the search was performed multiple times, each time with the term "Graph Theory" used as the fixed keyword but combined with one of the following complementary terms: "Ecological networks", "Landscape Fragmentation", "Landscape connectivity", "Landscape ecology", "Urban landscape". Finally, the search results were sorted in the descending order of the publication year (from 2014 to 2021).

Selection Process
The eligible articles were selected through the three-phase process used by Mehring et al. (2019) framework. The diagram of this process is illustrated in Fig. 1. In the first phase, articles that had the term "graph theory" and one of the complementary keywords ("Ecological Networks", "Landscape Fragmentation", "Landscape Connectivity", "Landscape Ecology", "Urban Landscape") in their titles, abstracts, or keywords were listed (duplicate articles were discarded). The number of articles in this phase was 677. After sorting the articles in the order of their publication year, in the second phase, the texts of the articles were examined to be clear whether the articles in the text also deal with GT or not? Those articles that were not related to the field of interest and those that had not used GT were discarded. This was done to enhance the focus in selecting articles on the subject under study and avoiding mistakes. Because in some researches, although keywords are included in the title, abstract and keywords, they are not used in the research text. Bringing these researches in the results and analysis causes an error in the research and diverts it from the main goal. Eventually, a total number of 375 scientific articles were found to meet these criteria. In the third phase, the articles that had not used the graph-based indexes were discarded, leaving 237 cases, which had employed graph-based indexes for ecological network analysis.

Analysis
The grounded theory (GT) was used to reach a simpler and more organized categorization of indexes used in the articles Fig. 1 The process of selecting articles so that the results can be presented and discussed in separate segments. This method involves systematic data collection followed by systematic analysis and simplification of the collected contents (Glaser and Strauss 2017). The core idea of this method is to gather data about a specific subject through research and then transform the gathered data into concepts and categories through repeated comparison and refinement . In this study, the VOSviewer software was applied for data visualization where needed (Perianes-Rodriguez et al. 2016). It is worth to mention that VOSviewer benefits from a multidimensional scaling technique for clustering data and envisaging bibliometric networks (Waltman et al. 2010).
In the analysis of articles, the following were investigated: geographical distribution of studies by country and continent, employed indexes, main indexes, number and composition of employed indexes, novel indexes, ecosystem composition of the use of graph-based indexes, frequency of use of graph-based indexes by the number and type of species, frequency of use of graph-based indexes in different fields of ecology. Although, the relationship between the indicators is investigated using Spearman correlation.

Results and Discussion
The last decade has seen the growing importance of research on ecological networks, and, therefore, tools for use in such research have also been considered. One of the tools of ecological network analysis is GT. GT greatly facilitates the analysis of functional (Iswoyo et al. 2018) and structural (Koohafkan and Gibson 2018) connections of the components of ecological networks. This theory can be used to simplify the analysis of relationships in the study of different animals and different ecosystems. In this study, the scientific literature concerning this subject was systematically analyzed to provide a better insight into the geographical distribution (Mehring et al. 2019) of studies that take this approach, the indexes commonly used in these studies, and the ecosystems and species on which these studies have been conducted.
To clarify the issue, as will be discussed below, we first mentioned the domains that have been studied. Then, to better understand the subject, we divided the species based on behavioural states, because by understanding the behavioural states of animals, their communication network can be analyzed. Then, by introducing the indicators, we put the main focus on them, such as: What species are they used for, which indicators have been used together, what is the percentage of use of indicators in studies, what is the dynamics of GT in presenting new indicators. Finally, with a better understanding of the position and status of GT in ecological network analysis, we will map its future in this area.

Geographical Distribution of Studies
The use of the graph-based approach in ecological research does not have an even geographical distribution. This approach has been used most frequently in Europe and Asia, Fig. 2 The distribution of countries in terms of article publication followed by North America. Only a limited number of studies in other regions have used this approach.
The results showed that the articles using this approach were limited to 38 countries (Fig. 2). In this respect, China (23.08%), France (11.34%), and the United States (9.72%) had published the highest number of the studies. Other nations with a notable number of articles were Spain (6.07%), Canada (4.45), and Mexico (3.64%). Also, 4.86% of the studies have not been focused on any specific country.
The results also show that researchers from 51 countries have taken part in the reviewed studies. The highest participation rates were further related to the researchers from China, the United States, and France with 15.89%, 13.29%, and 10.11%, respectively.
The examination of the geographical distribution of the use of GT in the field of ecology showed that the method is more widely in Europe and North America and also more evenly used across these continents. In Asia, most articles had been conducted in East Asia, especially China, with the largest number of case studies (n = 57) in the world. What is interesting about these results is the higher concentration of studies conducted in developed countries, which can be attributed to the higher priority given to ecological sustainability.
These studies have been conducted to examine the extent of habitat communication as well as animal communication, so this indicates that the critical areas in this regard are African, Asian and South American countries. The importance of South America and Africa in biodiversity, as well as the lack of studies on these continents, is a wake-up call for researchers, politicians and the public. Given that in these areas, animal habitats are disappearing one after another, so these results can draw the attention of officials in these countries for more policy and investment on conservation and environmental connectivity, and also attract the use of new scientific methods in this field.

Ecosystem Composition of the Use of Graph-Based Indexes in the Field of Ecology
Ecological indexes can be used in the analyses of all types of ecosystems. As expected, these indexes had been mostly recruited in the study of terrestrial ecosystems (85.23%). In 11.39% of the articles reviewed, the indexes had been employed to study aquatic ecosystems. The rest of the studies (3.38%) had not focused on any specific ecosystem (Fig. 3).
In this regard, the results show that graph-based indexes have been used more frequently in terrestrial ecosystems. Naturally, this could be due to higher accessibility of terrestrial ecosystems, availability of data, ease of observation, and ease of access to target species in these ecosystems.

Frequency of Use of Graph-Based Indexes by the Number and Type of Species
The articles reviewed (n = 237) were categorized from two perspectives: (1) the number of species studied and (2) the  type of species examined. Approximately 86.50% of the articles had been performed on one species, 5.91% of the cases had reflected on two species, and 3.38%, 0.84%, and 0.42% of the studies had been conducted on 3, 4, and 5 species, respectively. The remaining 2.95% of the articles were general. Table 1 shows the number of single-species studies that were conducted on each type of species. Table 2 shows the number of studies performed on two, three, four and five of each type of species. The total number of biological domains in which the studies had been conducted was 270 (277 if the seven general studies were included). To provide a better understanding of the frequency of use of GT in the domains of ecological analysis, the distribution of its use in different domains is plotted in Table 1. As the results show, the most frequent domains are animal species (50.90%), plant species (35.74%), context (10.83%), and general (2.53%). The articles on animals were also divided into five groups, viz. birds (9.75%), reptiles and amphibians (9.75%), mammals (23.10%), insects (4.33%), and aquatic species (3.97%). The studies on plants were further split into two groups, namely, aquatic (0.72%) and terrestrial (35.02%).
It was found that although a large portion of studies has been performed on only one species, GT is flexible enough to be used in the analysis of ecological connectivity and connections in multiple species. For example, in a 2016 study by Naicker et al., this approach was used to study five species.
The results also showed that most studies have been focused on animal species. Plant species and context are next in this ranking. Also, four of the studies have been general. These results demonstrate the wide applicability of GT in a wide variety of ecological network analyses. However, it should be noted that apparently this approach is mostly used for terrestrial plants and mammals. Nevertheless, given the flexibility of this approach, one can expect that it will be used more frequently and in a wider variety of applications. For example, in a 2019 study by Mestre in Iberia, context and landscape fragmentation were analyzed by considering roads and linear infrastructure as links and residential areas as nodes. In another example, Niculae et al. (2017) attempted to use a graph-based approach to investigate the spread of invasive species. This shows that GT can be used not only in ecological network analyses aimed at increasing connectivity Studies were performed on two species * * * * but also in ecological network analyses with the purpose of decreasing connectivity or other aspects of ecological connections when necessary. Therefore, GT can be expected to find wider application in ecological research, including in the spread analysis of viruses and infectious diseases.

Indexes
In the articles reviewed (n = 237), a total number of 118 indexes had been used 485 times. In these studies, the indexes were used in two ways: (1) individually, (2) combined with other indexes. In terms of frequency of use, some indexes were used for many years, but some others were new and had not been used as frequently. Therefore, this issue was taken into account during the analysis.
The frequency analysis showed that the studies had used the graph-based indexes 485 times. In terms of frequency of use, the indexes were divided into three categories: (1) widely used, (2) intermittently used, and (3) rarely used. The indexes PC (Probability of Connectivity), IIC (Integral index of connectivity), and LCP (Least-cost-path algorithm) fall in the first category. The most widely used index was PC, in 21.24% of the articles. The indexes of IIC and LCP were also second and third in this ranking by 13.61% and 7.63%, respectively. The indexes of BC (Betweenness Centrality), EC (Equivalent Connectivity), NC (Number of Components), NL (Number of Links), β (Node/Line Ratio), CC (Closeness Centrality), and GD (Graph Density), γ (Connectivity Rate), α (Complexity), CW (Clustering Coefficient), ASPL (Average Shortest Path Length), AWF (Area-Weighted Flux), Dg (Node Degree), IF (Interaction Flux), HI (Harary Index), had been also used in 1.03-5.57% of the studies, falling in the second category. The third category included 100 indicators that the frequency of their use in studies is from one to four (Tables 3, 4).
In the articles reviewed, 118 graph-based indexes had been used 485 times in the ecological network analysis. The results show that the indexes PC, IIC, and LCP have the highest frequency of use in these studies. PC, IIC, and LCP can be considered the most popular indexes for research in the field of ecology. It should be noted that the less frequent use of other indexes is not because of their ineffectiveness, but rather because many of them have been introduced more recently and it will take some time for them to become widely recognized and established. Nevertheless, the frequent use of PC, IIC, and LCP suggests that these three indexes can be trusted to yield fairly reliable results in ecological analyses. There are also a small number of old indexes that have not been used as frequently.
One of the benefits of using GT in ecological analyses is that graph-based indexes can be used in combination with each other to study multiple aspects of an ecological network simultaneously. Out of the 237 articles reviewed in this work, 49 cases had used two indexes. Out of these 49 studies, 18 articles had recruited a combination of PC and IIC, suggesting that these two indexes could be applied together in research in the field of the ecological network analysis.
The studies were also examined from the perspective of the composition of species studied. This examination showed that 85.96% of the articles had been focused on a single species. As well, 5.90% of the studies (n = 14) had reflected on two species. Among them, four articles had been on mammal and reptile-amphibian species (Almenar et al. 2019;Chaput-Bardy et al. 2017;Drake et al. 2017;Modica et al. 2021), six studies had shed light on mammal and bird species (Han and Keeffe 2019;Huang et al. 2018;Martensen et al. 2017;Ocampo-Peñuela et al. 2020;Peng et al. 2021;Petsas et al. 2021), one study had been conducted on mammal and plant species (Ruppert et al. 2016), one study had investigated reptile-amphibian and bird species (Pietsch 2018), one study had been on plant and bird species (Han and Keeffe 2021), and one study had been focused on plant and insect species (Corro et al. 2019). Of all the articles reviewed, 3.37% of the cases had been focused on three species. Among these studies, four cases were on mammal, bird, and reptile-amphibian species (Albert et al. 2017;Meurant et al. 2018;Sahraoui et al. 2017;Tarabon et al. 2021), one had been conducted on insect, aquatic, and context species (Ishiyama et al. 2014), one had focused on context, bird, and reptile-amphibian (Xiu et al. 2017), one was on plant, bird, and insect species (Wolstenholme and Pedley 2021), and one had investigated plant, insect, and aquatic species . Two studies had also worked on four species, including bird, reptile-amphibian, mammal, and insect species (Bourgeois and Sahraoui 2020; Tarabon et al. 2020). There was also one study on five species, including plants, birds, reptiles and amphibians, mammals, and insects (Naicker et al. 2016).
Each year, researchers introduce new graph-based indexes for ecological research. For example, in 2017, 14 new indexes, and in 2018, 15 new indexes were proposed in the field of the ecological network analysis. Surprisingly, 28 new indexes were introduced in 2021.
This shows the dynamic nature of the application of GT in the field of ecology. Since the low usage frequency of some of these indexes is related to their novelty, it is still too soon to judge their effectiveness on this basis. Therefore, the usage frequency investigation should be repeated at a later date to see which indexes emerge as strong and popular options for use in ecological analyses compared to the years before (Period 2014-2021).   (2021)

PC
The PC index was used in 103 articles on 113 samples and has a frequency of approximately 21.24% among the indicators used. PC is the most popular index in this field, as researchers have used it incessantly over many years. This shows the effectiveness and functional validity of this index in connectivity analyses and its high acceptance among researchers in this field. PC has been used both individually and in combination with other indexes, which shows the flexibility of this index and its descriptive power when combined with other indexes. PC has been most commonly used in the ecological analysis of plants and mammals, and especially the latter. This shows that PC is more reliable for use in the analysis of connectivity in these fields. Of course, PC has been used in other areas as well and the results suggest that it can be used to analyze ecological networks of different varieties. The PC index was used to analyze the ecological network in all categories of plants, birds, reptiles and amphibians, insects, aquatic animals, and context. However, it has been used most in mammals and plants. This index is a leading indicator among GT indicators in the field of habitat network analysis. The high use and diversity of the studied species show its ability. However, it should be borne in mind that the PC index has a long history of ecological chip analysis, its low use in aquatic species with the frequency of 1 (Appol-  Zhou et al. 2021), It will raise the question of whether it has capabilities in the field of aquatic species, insects and the field or not? Therefore, more caution or research is needed on the mentioned species to prove its capability.

IIC
The IIC index had been recruited in 66 studies, with 13.61% of the total 118 indexes, it has the second place in the use of GT indices in ecological network analysis. Although the IIC index has been used in all categories performed for ecological network analysis. The most use of this index is in the field of plants, so that out of 74 uses of this index in 39 cases has been used to analyze connections in plant fields. Therefore, the use of IIC index for analysis of connections in plant domains has more reliability and validity. This index has been used in other fields as well. It has been used as a mammalian analysis of connections with a frequency of 14. The use of the index in different contexts indicates its flexibility and application. However, the low use of the IIC index in the domains of amphibians and reptiles (Almenar et al. 2019;Matos et al. 2019;Modica et al. 2021;Naicker et al. 2016), birds (Goulart et al. 2015;Han and Keeffe 2019;Naicker et al. 2016), aquatic animals (Ishiyama et al. 2014(Ishiyama et al. , 2015(Ishiyama et al. , 2020, and insects (Giannini et al. 2015;Ishiyama et al. 2014;Naicker et al. 2016;Paterson et al. 2019) raise the question of whether it can analyze the ecological network in these domains as well.

LCP
LCP has been used in 37 of the reviewed studies and has a usage frequency of 7.63%, which makes it the third most frequently used graph-based index in ecological analyses. This index has been used 47 times in various fields. Unlike Transboundary connectivity TC 1 Santini et al. (2016) the previous two indexes, LCP has been used alone in all the studies. This index has been used in all categories of analysis except the analysis of insect species. The interesting point regarding LCP, however, is its frequent use in the analysis of connectivity for reptile-amphibian species with frequency 9, which shows the suitability of this index rather than IIC for use in this particular field. In comparison to IIC and PC, LCP has a wider area of application.

BC and EC
The EC index with the frequency of 5.57% had been employed in 27 articles. The remarkable thing about this index was that 35 species had been studied, wherein mammals with the frequency of 13 were in the first place and plants with the frequency of eight were in the second place.
In relation to the EC index, it should be added that the lowest use of this index was for context (Mu et al. 2021) with the frequency of one but not used in aquatic animals. The BC index had been also utilized in 26 studies. The highest use of this index was for plants and mammals with the frequency of seven and eight, and the lowest was for insects (Heer et al. 2021) and birds (Petsas et al. 2021) with the frequency of one.

Other Indexes
NC and NL were among the most considerable indicators with 19 and 11 times use, respectively. Plant species had been also the subject of articles using these indicators. In relation to NC, 12 out of 19 articles had worked on plant species, while mammals (Gao et al. 2017;Modica et al. 2021), reptiles and amphibians (Godet and Clauzel 2021; Modica et al. 2021), insects (Heer et al. 2021;Paterson et al. 2019), and context (Mu et al. 2021;, had the frequency of two. As well, no studies had used the NC index in relation to aquatic animal and bird species. Considering the NL index, 8 out of 11 studies had been performed on plant species, while the frequency of the articles on other species was one (namely, mammal, reptile and amphibian (Modica et al. 2021), insect (Heer et al. 2021), and context . The rest of the indexes had the frequency of less than 10, as follows: the frequency of six for the β, CC, GD, ASPL, and AWF indexes, the frequency of five for the α, CW, Dg, γ, IF, and HI indexes, the frequency of four for the MST, F, NN, and CD indexes, and the frequency of three for the DC, ANC, GDi, and MCR indexes. Finally, eight and 84 indicators had the frequency of two and one, respectively.

Statistical Relationships Between Indexes
To explore the relationship between the indexes, the Pearson correlation is conducted. Because the main indexes frequency was less than two, any statistical analysis was meaningless. Therefore, the correlation was calculated for indexes with equal or more than two frequencies. According to the study results in Table 5, a limited specific relationship was considerable. Here, only five significant positive relationships were observed between the indexes. The γ and α indexes also showed the strongest relationship (+ 1), and then the correlations of α and β (0.841), γ and β (0.841), CW and APSL (0.606), and finally. Table 5 depicts the Pearson's correlation matrix and Table 6 portrays the significance levels.
The similarities of the indexes and the correlations between them could appear in a cluster of related indicators. The highest correlation of the α and γ indexes shows their very high relationships in terms of the evaluation of the ecological networks. The α index was also used to indicate the degree of network closure and characterize the degree of loop occurrence in the network and the γ index reflected the measured connectivity degree of all nodes in the network (Huang et al. 2021a;Nie et al. 2021;Zhao et al. 2019). Both of these indicators had a high correlation (0.841) with the β index. The β index was thus assumed as a simple metric for network complexity (Zhao et al. 2019). The three indexes, α, β, and γ had been also used together to quantitatively evaluate ecological networks and quantitatively compare the status of the network with its ideal conditions.
Another obvious correlation was observed between CW and APSL with a value of 0.606. The CW index measured the average fraction of the patch's neighbors (Guzmán-Colón et al., 2020), and the ASPL was the average length of the shortest path connecting the patch pairs in the ecological network (Heer et al. 2021). The lower the mean of the ASPL, the higher the connectivity and the coherence between the pairs of nodes, and the higher the CW among the adjacent nodes.
Despite less statistical similarities, the PC, IIC, and LCP have the main used indexes. Moreover, the used indicators could be classified into three categories: first those indicators which are used lonely (frequency = 132). This group includes 55.69% of the studies that have used only one index; second, 20.80% of studies have used two indexes (frequency = 47); third, 8.41% used three indexes (frequency = 19). In the Last group, a combination of more than four indicators (frequency = 30) ( Table 7). The highest number of indexes used in a study is fifteen, which has occurred in only 0.42% of the studies.
As mentioned, the indexes most commonly used in the studies are PC and IIC with usage frequency of 21.24 and 13.61%, respectively. The results show that in 24.47% of the studies, PC has been used in combination with other indexes. For IIC, this figure is 19.40%. This shows the popularity of combining PC and IIC with other indexes. Considering the years in which these indexes have been used in combination with others, it can be concluded that they have not lost their analytical value over time.
Dynamics of GT indicators given the dynamic nature of GT-based analyses in the field of ecology; researchers introduce new indexes for this field every year. In this study, the 2014 figures were considered as baseline. Table 8 shows the number of graph-based indexes introduced in the field of ecology in each year since 2014. In 2018, for example, a   Table 6 .

Identification Major Gap
Resilience is a very important issue in the field of ecological networks. It can be defined as maintaining the structure, functioning and internal feedback of ecological networks in the face of threats (Holling 1973(Holling , 1996Maia et al. 2021). Although, resilience was mentioned in a few studies theoretically and mainly in the introduction. Nevertheless, it has not been seriously studied in the context of GT. Similarly, no index is provided for analysis of the ecological network's resilience in the GT milieu. However, various theories have been proposed regarding ecological network resilience including tragedy of the commons (TOC) (Hardin 1968), paradox of enrichment (POE) (McCann et al. 2021), paradox of redundancy (POR), and competitive exclusion principle (CEP) (Lewis 2019). It is worth mentioning that, ecological network resilience forms one of the critical filed of studies in general, while it performs a significant gap in the environment of GT. It is thought-provoking when it considers from a theoretical view of point. The TOC theory describes competition over share resources in which a shared resource for  One indicator  132  PC, IIC, LCP, BC, DNMS, ED, DF, SDM, ASPL, IM, MSSN, HI, CAT, SRM, EC, GD, KP, CT, MPG,  RNG, PMC, MCA, BM, HI, GCC, MLT, VIF, BEI, TOFCL, MST, GED, LCC, IF, MCR, FCM, CD,  MST, DMC  Two indicators  49  PC-IIC, PC-EC, STRG-GED, LCD-RD, PC-BC, CI-LI, ASPL-FBC, SPT-LCP, IIC-LCP, SHI-Fi, IIC-NC, IIC  several species or individuals of the same species is lost or becomes more difficult to access. In this case, this resource is only available to members who are in a better position, while its negative effects are shared by all members (Roopnarine 2013). As this process continues, everyone is forced to increase their use of the resource, and this overuse consumes the resource and destroys its resilience. Finally, it exposes the ecological network to fragmentation and destruction. As result, some novel changes will be emerged in the configuration of the graph and it could be the subject of new indexes. In POE framework, the paradox of enrichment is related to the abundance of prey food in relation to the diversity stability of predators. For example, increasing the food of a prey increases it, and as a result, the abundance of prey increases the number of species of hunter that are more capable of hunting that species. Eventually, the increase in the abundance of one species of hunter causes the elimination of other hunters, which leads to the imbalance of the ecosystem or the environment (Rana et al. 2013). The network changes will be the result of the imbalance in the ecosystem that shapes a considerable subject in the evaluation of ecological networks resilience in GT context. POR theory states that as the number of components increases and the network expands, if a problem occurs in their common feature, the probability of catastrophic failure occurs throughout the network increased (Berniker and Wolf 2001). This theory considers another aspect of network challenges and large-scale connectivity as a threat (Peterson et al. 2017). So that if a transferable problem occurs in one of the members, due to the communication of the whole system with each other, it is more likely to spread to all members of the network and as a result, it may destroy the whole network. As result, the mentioned challenge could play a critical role in the resilience of GT.
Furthermore, CEP states that two species with identical niches cannot coexist indefinitely and finally, one of them is removed (Kneitel 2019;Mueller 2019). Another way of expressing this principle is to pay attention to the fact that for unlimited coexistence, different species must have distinct ecology (Holt 2017). Otherwise, their stability will not continue and it is likely that by deleting one of the species, their environment will change or the environmental balance will be lost. The result will be reflected in the ecological network.
As can be seen in each of these theories, they all examine the relationships of different species in relation to ecological resilience as members of one species or several species. These connections can be analyzed and examined as a network. As mentioned earlier, given that g GT provides the structure and indicators to evaluate ecological networks, it can be very useful and used in connection with the study of ecological networks that use these theories to study and analyze the resilience of ecological networks.

Conclusions
In this study, the past studies that have used GT in the analyses of ecological networks were reviewed. In general, it was found that the GT is flexible enough for the analysis of different ecological networks and has been extensively used in recent years to study and analyze ecological connections and connectivity in different ecosystems and a wide variety of species. For these reasons, GT is useful in the analysis of the ecological network of habitat networks. Therefore, in the future, this approach can be expected to be used more widely in the area of ecological network analysis.
One of the features of the use of GT in ecological network analyses is the continuous emergence of novel indexes, as every year researchers develop several new graph-based indexes for use in these analyses. Therefore, one can conclude that GT has enough flexibility to answer questions that may arise in the future in the field of ecological network analysis.
Although researchers are introducing novel graph-based indexes for ecological network analyses every year, the credibility of many of these indexes will be determined in time by whether they will be effectively used in future researches. But so far, PC, IIC, and LCP seem to be the most credible graph-based indexes for use in ecological network research. The absent aspect in studies conducted using GT in the field of ecological network analysis is the resilience of ecological networks. Unfortunately, so far, less serious has been presented that examines the resilience of ecological networks using GT.
There are many advantages to the use of GT in ecological network analysis as it has low information requirements and can be used in research on different species. Therefore, researchers need to direct more attention and effort to the development and adaptation of this theory for use in ecological network analyses. It is hoped and expected that researchers will make further use of this theory in the analyses of ecological networks, expand it for use on different aspects of connectivity in different species, and use it to develop more accurate and useful tools and indexes for these analyses.