Graph clustering via generalized colorings

Redundancy is a typical characteristic of complex networks. In ecological networks, redundancy also exists in the connectivity between key nodes. Understanding the redundant structure of an ecological network and identifying the backbone structure is very important in maintaining the stability of the ecological network system. Using Shenzhen City as an example, we aimed to construct an ecological network and identify its arterial corridors. A protection plan for arterial corridors is proposed through network analysis. Using the concept of edge betweenness and a minimum spanning tree, we identified the arterial corridors in the ecological network. Four corridor destruction scenarios were considered: random destruction, urban sprawl, ecological control line, and arterial corridor scenarios. The ecological network stability under the four scenarios is explored based on a network robustness analysis. The ecological network covers 32 important habitats and 49 ecological corridors. Among them, 31 arterial corridors connect 32 important habitats, forming the backbone structure of the ecological network. Both the control line policy and protection plan of the arterial corridor play a role in maintaining the stability of the ecological network. The stability of the ecological network under the arterial corridor protection scheme was 6% higher than that under the control line one. This study provides an effective method for identifying arterial corridors in an ecological network. Protecting the arterial corridors and the redundant corridors based on edge betweenness can maintain the stability of the ecological network to the greatest extent. Our study has reference significance for the hierarchical protection of ecological corridors.

A generalization of a theorem of de Bruijn and Erdös on the chromatic number of infinite graphs

Given an arbitrary graph, wh~t can we say about the setl of all chromatic numbers of its subgraphs ? In particular, for e~rdinals m ~ n, does every graph of chromatic number m have a subgraph of chromatic number n? For n ~ ~t 0 the affirmative answer follows from the theorem of DE BI~UIJ~ and ERD6S [1], SO the simplest nontrivial question is whether every graph of chromatic number ~2 has a subgraph of chromatic number Nr I have no positive results; the purpose of this note is only to state the probleml in the form of 2 dubious conjectures, and to prove that the stronger conjecture implies a weak form of the GCH. If % is a set, [S] 2 = {X c~ S " [X l = 2}. We regard a graph as a system G = (V, E) consisting of a set V of vertices and a set E ~c [ V] ~ of edges. A coloring of G is a function f defined on V such thatf(x) :~f(y) whenever {x, y} # E. G is m-colorable if there is a coloring f such that If(V) l ~ m. The chromatic number of G, ehr(G), is the least m such that G is m-eolorable. A graph G" ---- (V', E') is a vertex subgraph of G if V' ~ V and E' = E N [V'~]; an edge subgraph if V' ~ V and E' c E. ConJEcturE 1. For m ~ n, every graph of chromatic number m has a vertex subgraph of chromatic number n.

Rapid urbanization has greatly altered urban ecological spaces and habitat quality functions, threatening regional biodiversity and the sustainability of landscapes. Therefore, constructing a comprehensive ecological network and ecological safety patterns is crucial for ecosystem management and regional development. However, simple quantification of ecological networks fails to meet the construction needs of ecological safety patterns, and most studies focus solely on network quantification analysis, thus overlooking the importance of spatial analysis. This study proposes a method of ecological network quantification assessment combined with hotspot analysis and coupled with standard deviational ellipse spatial analysis, which not only satisfies quantitative analysis but also adds spatial analysis methods, facilitating a more comprehensive construction of safety patterns. Firstly, through morphological spatial pattern analysis (MSPA) and landscape connectivity indices, ecological source areas in the main urban area of Kunming were identified, integrating various resistance factors and corrective factors to construct an ecological resistance surface. The minimum cumulative resistance (MCR) model was used to identify potential ecological corridors, and their importance was evaluated using the gravity model, thus establishing an ecological network. Secondly, based on network structure indices, the ecological network was assessed and optimized. On this basis, combined with hotspot analysis coupled with standard deviational ellipse spatial analysis, an ecological safety pattern was constructed. The results show the following: the core area of the study region is 2402.28 km2, accounting for 52.07% of the total area; there are 13 ecological source areas, totaling 2102.89 km2, accounting for 45.58% of the total area; there are 178 potential ecological corridors, including 15 level-one ecological corridors and 19 level-two ecological corridors; and 103 ecological nodes, 70 “stepping stones”, and 48 ecological breakpoints were identified. In terms of ecological network optimization, six new ecological source areas were added, covering an area of 16.22 km2, and the potential ecological corridors increased to 324, including 11 new level two ecological corridors, 51 new ecological nodes, 15 “stepping stones”, and 24 major ecological breakpoints. After optimization, the network closure index (α), network connectivity index (β), and network connectivity rate index (γ) improved by 15.16%, 24.56%, and 17.79%, respectively. Based on the network structure quantitative analysis and hotspot analysis coupled with the standard deviational ellipse’s spatial analysis, a “one axis, two belts, five zones” ecological safety pattern was constructed.

The conservation and renovation of biological diversity of territories are of vital importance in Ukraine. The concept of ecological networks has laid the foundation of these issues. In Ukraine, the law “On the ecological network” was adopted, and ecological networks were developed at the regional level. However, most developments do not fully take into account the location of reserved territories and places of concentration of valuable flora and fauna, and also often do not correspond to the principle of formation of a holistic system, which significantly reduces their environmental significance and the expediency of practical implementation. The authors of the article consider that one should estimate the natural potential of the territories in detail and objectively, as well as provide recommendations for practical actions for the renovation of biological and landscape diversity. One can do it in the process of the development of local ecological networks (at the level of separate administrative regions and territorial communities). One has developed the structure of the ecological network on the example of Stryi district of Lviv region and determined the estimation of the spatial connectivity of its crucial elements. The problem is that most of the connecting territories are formed from semi-natural and unnatural lands (hayfields, pastures, arable lands), which complicates the practical side of the project implementation. One needs to develop an effective mechanism to encourage landowners and land users to carry out environmental activities; to increase investments in re-naturalization of lands; to conduct environmental education and environmental public awareness campaign; to provide local government and other stakeholders with consulting services for the development of ecological network development programs at the local level in Ukraine.

Quantifying the structure and dynamics of species interactions in ecological communities is fundamental to studying ecology and evolution. While there are numerous approaches to analysing ecological networks, there is not yet an approach that can (1) quantify dissimilarity in the global structure of ecological networks that range from identical species and interaction composition to zero shared species or interactions and (2) map species between such networks while incorporating additional ecological information, such as species traits or abundances. To address these challenges, we introduce the use of optimal transport distances to quantify ecological network dissimilarity and functionally equivalent species between networks. Specifically, we describe the Gromov–Wasserstein (GW) and Fused Gromov–Wasserstein (FGW) distances. We apply these optimal transport methods to synthetic and empirical data, using mammal food webs throughout sub‐Saharan Africa for illustration. We showcase the application of GW and FGW distances to identify the most functionally similar species between food webs, incorporate additional trait information into network comparisons and quantify food web dissimilarity among geographic regions. Our results demonstrate that GW and FGW distances can effectively differentiate ecological networks based on their topological structure while identifying functionally equivalent species, even when networks have different species. The FGW distance further improves node mapping for basal species by incorporating node‐level traits. We show that these methods allow for a more nuanced understanding of the topological similarities in food web networks among geographic regions compared to an alternative measure of network dissimilarity based on species identities. Optimal transport distances offer a new approach for quantifying functional equivalence between networks and a measure of network dissimilarity suitable for a broader range of uses than existing approaches. OT methods can be harnessed to analyse ecological networks at large spatial scales and compare networks among ecosystems, realms or taxa. Optimal transport‐based distances, therefore, provide a powerful tool for analysing ecological networks with great potential to advance our understanding of ecological community structure and dynamics in a changing world.

The accelerated development of urban agglomerations in China has resulted in the significant regional expansion of infrastructure and urban spaces, which has led to the fragmentation of habitats and the degradation of ecosystem function. Ecological networks have been shown to reconnect isolated habitat patches within urban agglomerations by identifying ecological sources and constructing corridors, which could enhance regional ecological security. Nature reserves, as critical areas for the protection of key species and ecosystems, play a vital role in this process. Investigating the influence of nature reserves on the ecological networks of urban agglomerations helps to integrate regional ecological resources, optimize ecological network structures, and enhance cross-departmental coordination in nature reserve management and ecological environment protection. Using the urban agglomeration in the middle reaches of the Yangtze River as a case study, this research analyzes the impact of nature reserves on the ecological network of urban agglomerations. Initially, ecological source patches are identified using the “Quality-Morphology-Connectivity” evaluation model. Different types of nature reserves are then superimposed to create four distinct source schemes. Subsequently, a resistance surface is constructed through a comprehensive evaluation method to assess ecological barriers. Then, ecological corridors are generated using circuit theory tools. Finally, a comparison of the effectiveness of the four ecological networks is conducted using 12 landscape pattern metrics. The results indicate several key points. Firstly, the inclusion of nature reserves is shown to supplement ecological sources and increase corridor numbers, thereby enhancing the optimization effect of the urban agglomerations’ ecological network threefold. Secondly, the impact of nature reserves on the ecological network is closely related to the spatial scale of patches, and patch scale consistency should be considered to prevent network functionality loss. Thirdly, establishing a cross-departmental and cross-regional collaborative management mechanism is recommended to organically integrate nature reserves with ecological networks. These results provide a data-driven foundation for the optimization of ecological networks in urban agglomerations and inform effective management strategies for nature reserves, to promote the construction of ecological civilization in urban agglomerations.

Knowledge of species composition and their interactions, in the form of interaction networks, is required to understand processes shaping their distribution over time and space. As such, comparing ecological networks along environmental gradients represents a promising new research avenue to understand the organization of life. Variation in the position and intensity of links within networks along environmental gradients may be driven by turnover in species composition, by variation in species abundances and by abiotic influences on species interactions. While investigating changes in species composition has a long tradition, so far only a limited number of studies have examined changes in species interactions between networks, often with differing approaches. Here, we review studies investigating variation in network structures along environmental gradients, highlighting how methodological decisions about standardization can influence their conclusions. Due to their complexity, variation among ecological networks is frequently studied using properties that summarize the distribution or topology of interactions such as number of links, connectance, or modularity. These properties can either be compared directly or using a procedure of standardization. While measures of network structure can be directly related to changes along environmental gradients, standardization is frequently used to facilitate interpretation of variation in network properties by controlling for some co-variables, or via null models. Null models allow comparing the deviation of empirical networks from random expectations and are expected to provide a more mechanistic understanding of the factors shaping ecological networks when they are coupled with functional traits. As an illustration, we compare approaches to quantify the role of trait matching in driving the structure of plant-hummingbird mutualistic networks, i.e. a direct comparison, standardized by null models and hypothesis-based metaweb. Overall, our analysis warns against a comparison of studies that rely on distinct forms of standardization, as they are likely to highlight different signals. Fostering a better understanding of the analytical tools available and the signal they detect will help produce deeper insights into how and why ecological networks vary along environmental gradients.

• The study shows that the ecological source area in Ningwu County nearly doubled from 2000 to 2020, indicating the effectiveness of ecological restoration programs. • With the emergence of new ecological source patches, the number of ecological corridors increased from 8 to 20, significantly enhancing the connectivity of the ecological network. • Characterized ecological network evolution in a county where a series of ecological programs were implemented, demonstrating effective enhancement of ecosystem integrity and connectivity. Ecological network construction has been conducted in various areas worldwide at different spatial scales, to guide regional ecological conservation and restoration, or predict future ecological networks under different land use scenarios. However, it has been rarely applied to evaluate the ecological restoration achievements from the aspect of ecosystem integrity and connectivity. This study uses MSPA-MCR model to characterize ecological network changes before and after a series of ecological restoration programs were implemented in Ningwu County, China. The results showed that: (1) The ecological source area first grown slightly from 2000 (344.51 km 2 ) to 2010 (353.49 km 2 ). And then the number almost doubled in 2020 (614.39 km 2 ). With the expansion of the area, the original 6 ecological source patches merged into 4 in 2010 and then 3 in 2020, and 1 new patch emerged in 2010 and 5 more patches in 2020. (2) The area of high ecological resistance showed a minor increase from 2000 (576.90 km 2 ) to 2010 (592.31 km 2 ), but then declined sharply by 38.53 % in 2020 (364.07 km 2 ). (3) With the emergence of new ecological source patches, more corridors were needed to connect them. The ecological corridors increased from 8 in 2000 to 9 in 2010, and 20 in 2020; and the total length increased from 70.23 km in 2000 to 144.23 km in 2010, and 170.21 km in 2020. (4) The ecological network topological structure indices showed that the ecological network structure improved over the two decades. These results indicate that the series of ecological restoration programs effectively enhanced the ecosystem integrity and connectivity. Further analysis suggests that the substantial increase of ecological source area was due to the ecosystem service enhancement on existing ecological land and the emergence of new planted forest land. And implications for future ecological restoration were given based on the ecological network structure.

The rapid urbanization intensifies the conflict between land use and ecological protection. While existing studies acknowledge the role of ecological networks in mitigating human-land conflicts, most focus on one-way impact analyses and neglect the dynamic feedback mechanisms between land use change and ecological network evolution. This study develops a bidirectional feedback framework integrating ecological networks and land use simulation. By quantifying the ecological network's constraint effect, we adjust the development probabilities in land use simulations, update predicted land use patterns, and synchronously update the ecological network structure. Through iterative feedback, we identify core areas critical to regional ecological security. Results reveal a strong feedback mechanism between land use change and ecological network structure, with spatially differentiated mutual influences. The bidirectional feedback suppresses construction land expansion near ecological networks, redirecting development to distant areas. Applying the framework to Huanggang City, we identify 2257.55 km2 of key ecological source areas and 20.56 km2 of core ecological corridors as future ecological security core areas (FESCA). This study advances dynamic feedback analysis of ecological network-land use co-evolution, offering scientific insights for regional spatial planning and ecological conservation.

We propose a model of flow networks based on information coding and channel to discuss the stability. To study the relation between the structure and the stability of networks, we regard a flow network as an information source, and perturbation from the environment as disturbance in channel. We define an information index (H2-information) and the related indeces to evaluate the structure of networks. H2-information shows the degree of organization of the network. We apply the proposed information indices to ecological systems and study some properties between the stability and the structure of ecological flow networks.

Using network ecology to understand and mitigate long‐term insect declines

Constructing a rational ecological network is crucial for balancing regional development with environmental protection. However, existing research typically emphasizes the analysis of overall patterns, lacking an in-depth exploration of the dynamic changes in key elements and the interactions between different components. Using the Yongding River floodplain as a case study, this study applied morphological spatial pattern analysis, landscape connectivity metrics, and biodiversity assessments to identify core ecological source areas. Circuit theory was used to delineate ecological corridors and analyze network evolution across four key years, while graph theory facilitated an in-depth analysis of network structural characteristics. Furthermore, key areas for ecological restoration were identified within the floodplain. We found that the number of ecological source patches in the study area has remained relatively stable, though their total area has shown a fluctuating decline, accounting for approximately 10% of the floodplain. Additionally, ecological corridors have decreased significantly from 1967 to 2021, with a marked reduction in major corridors, leading to increased resistance to material and energy flow and a corresponding decline in network connectivity and stability. More importantly, current ecological pinch points are primarily distributed in a bead-like pattern along the Yongding River channel, while ecological barriers are concentrated in the northern and eastern floodplain, often at intersections of dense road networks and ecological corridors. These critical areas of fragmentation within the ecological network are prioritized for targeted ecological protection and restoration efforts. Overall, this study advances our understanding of the spatial distribution and composition of key ecological elements within river corridor networks and offers a framework for evaluating these networks through a multidimensional optimization approach for ecological source patches. At the same time, we conducted an in-depth analysis of key fragmentation areas in the Yongding River floodplain, providing valuable guidance for future ecological protection and restoration initiatives in river corridors.

Cyclic dependence, vertical integration, and innovation: The case of Japanese electronics sector in the 1990s

Constructing ecological networks (ENs) is an effective measure to mitigate the conflict between urban development and ecological conservation. However, existing simulating methods lack adequate consideration of human ecological demands and the spatial scale differences between these demands and natural ecological processes. This might lead to issues such as incomplete ecological process cycles or structural mismatches being overlooked during ENs simulations. To address these gaps, this study proposed an urban multi-scale nested ENs simulating framework that integrates human ecological demands with natural ecological processes. The framework first simulated an ENs focused on natural ecological process cycles at a global scale (GS). Then, it simulated an ENs centered on human ecological needs within the core urban areas at local scale (LS). Finally, it nested these multi-scale ENs by using cross-scale ecological supply sources as connecting points. This framework was applied to simulate spatio-temporal pattern changes in ENs of Jinan City, a core city in downstream of the Yellow River in China, aiming to mitigate cross-scale ecological conflicts between human–nature interactions under the background of urbanization. The study’s findings revealed that the area of demand sources increased by 8.56 times over 20 years. the area of cross-scale supply sources decreased by 15 km2 relative to 2000, and the deterioration in connectivity was more pronounced in GS compared to LS, with a decline of approximately 13.8%. These changes indicate the presence of incomplete ecological process cycles and structural mismatches across the multi-scale boundaries within the study area, which have been worsening annually. We recommend optimizing Jinan City’s multi-scale ecological network through three key strategies: rectifying internal structural mismatches, protecting core ecological areas, and aligning regional ecological demands. Implementing these strategies could significantly improve the network structure, reduce cross-scale mismatches, and enhance ecological connectivity by about 9%. This study highlights the importance of addressing structural mismatches and promoting complete ecological cycles in urban multi-scale ENs simulating, providing valuable insights for formulating urban multi-scale ecological conservation and restoration policies.