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Abstract The explosive growth of the Web and of social networks motivates the need for analyzing the macroscopic structure of their underlying graphs. Although the characterization of the structure of a graph with respect to its pairwise connectivity has been known for over 15 years, just one subsequent study analyzed the world inside the giant strongly connected component, where it has been shown that the largest strongly connected component has its own microscopic bow-tie structure defined with respect to pairwise 2-connectivity among its vertices. In this paper, we introduce the local microscopic bow-tie structure of the largest strongly connected component, demonstrating its self-similarity property. Our experiments, conducted on the several Web graphs and social networks demonstrate clear structural differences between considered Web and social networks.

Objective: Surveillance is critical for the rapid implementation of control measures for diseases caused by aerially dispersed plant pathogens, but such programs can be resource-intensive, especially for epidemics caused by long-distance dispersed pathogens. The current cucurbit downy mildew platform for monitoring, predicting and communicating the risk of disease spread in the United States is expensive to maintain. In this study, we focused on identifying sites critical for surveillance and treatment in an attempt to reduce disease monitoring costs and determine where control may be applied to mitigate the risk of disease spread. Methods: Static networks were constructed based on the distance between fields, while dynamic networks were constructed based on the distance between fields and wind speed and direction, using disease data collected from 2008 to 2016. Three strategies were used to identify highly connected field sites. First, the probability of pathogen transmission between nodes and the probability of node infection were modeled over a discrete weekly time step within an epidemic year. Second, nodes identified as important were selectively removed from networks and the probability of node infection was recalculated in each epidemic year. Third, the recurring patterns of node infection were analyzed across epidemic years. Results: Static networks exhibited scale-free properties where the node degree followed a power-law distribution. Betweenness centrality was the most useful metric for identifying important nodes within the networks that were associated with disease transmission and prediction. Based on betweenness centrality, field sites in Maryland, North Carolina, Ohio, South Carolina and Virginia were the most central in the disease network across epidemic years. Removing field sites identified as important limited the predicted risk of disease spread based on the dynamic network model. Conclusions: Combining the dynamic network model and centrality metrics facilitated the identification of highly connected fields in the southeastern United States and the mid-Atlantic region. These highly connected sites may be used to inform surveillance and strategies for controlling cucurbit downy mildew in the eastern United States. [ABSTRACT FROM AUTHOR]

Objective Surveillance is critical for the rapid implementation of control measures for diseases caused by aerially dispersed plant pathogens, but such programs can be resource-intensive, especially for epidemics caused by long-distance dispersed pathogens. The current cucurbit downy mildew platform for monitoring, predicting and communicating the risk of disease spread in the United States is expensive to maintain. In this study, we focused on identifying sites critical for surveillance and treatment in an attempt to reduce disease monitoring costs and determine where control may be applied to mitigate the risk of disease spread. Methods Static networks were constructed based on the distance between fields, while dynamic networks were constructed based on the distance between fields and wind speed and direction, using disease data collected from 2008 to 2016. Three strategies were used to identify highly connected field sites. First, the probability of pathogen transmission between nodes and the probability of node infection were modeled over a discrete weekly time step within an epidemic year. Second, nodes identified as important were selectively removed from networks and the probability of node infection was recalculated in each epidemic year. Third, the recurring patterns of node infection were analyzed across epidemic years. Results Static networks exhibited scale-free properties where the node degree followed a power-law distribution. Betweenness centrality was the most useful metric for identifying important nodes within the networks that were associated with disease transmission and prediction. Based on betweenness centrality, field sites in Maryland, North Carolina, Ohio, South Carolina and Virginia were the most central in the disease network across epidemic years. Removing field sites identified as important limited the predicted risk of disease spread based on the dynamic network model. Conclusions Combining the dynamic network model and centrality metrics facilitated the identification of highly connected fields in the southeastern United States and the mid-Atlantic region. These highly connected sites may be used to inform surveillance and strategies for controlling cucurbit downy mildew in the eastern United States.

Over the last decade, we have witnessed the emergence of numerous online social media platforms and a growing research interest in understanding their underlying structures and mechanisms. This paper investigates the formation and evolution of the EcoNation social media platform, which is designed to foster collaboration for environmental sustainability.

Introduction: The psychometric properties of the Satisfaction With Life Scale (SWLS) have been evaluated across numerous languages and population groups, primarily from a factor analysis perspective. In some studies, inconsistencies in structural invariance have been identified. Objective: This study aims to analyze the properties and gender invariance of the SWLS from a network analysis perspective. Method: A total of 857 Spanish university students were obtained through a stratified random cluster sampling method in a cross-sectional survey design study. Descriptive analysis of the items, partial-correlation network, Bayesian network model estimation, and invariance analysis by gender were conducted. Results: The instrument did not exhibit any floor or ceiling effects. Each item can be considered univariately normally distributed, and all items clustered in a single and stable community. The partial-correlation network model and centrality measures were stable in the full sample and invariant across genders. Item 3 emerged as the most central node in the network with the highest predictability. The Bayesian network indicated that items 2 and 4 initiate the process, while item 5 acts as the sink, and items 1 and 3 act as mediators. Conclusions: The SWLS can be used as a unidimensional measure, and the total score and relationships among items are stable and reliable. Any potential differences among genders cannot be associated with the functioning of the instrument. The predictability of every item was high, and the Bayesian network clearly identified different roles among the items. [ABSTRACT FROM AUTHOR]

This study investigates the research questions: "How do political connections within Greece's governing party evolve, and what underlying patterns and dynamics are revealed through a network analysis of interactions on  X  (formerly Twitter)?" To address these questions, data were collected from X , focusing on following, retweeting, and mentioning activities among the politicians within the governing party. The interactions were meticulously analysed using tools derived from Network Theory in mathematics, including in and out-strength centrality, hubs and authorities centralities, and in and out-vertex entropy. In line with the emerging field of data journalism, this approach enhances the rigour and depth of analysis, facilitating a more nuanced understanding of complex political landscapes. The findings reveal complex and dynamic structures that may reflect internal relationships, communication strategies, and the influence of recurring events on these connections within the party. This study thus provides novel insights into understanding political communication via social networks and demonstrates the applicative potential of Network Theory and data journalism techniques in social sciences. [ABSTRACT FROM AUTHOR]

Abstract Emerging in the twenty-first century, Network Science provides practical measures to interpret a system’s interactions between the components and their links. Literature has focused on countries’ interconnections on the final goods, but its application on the value-added from a network perspective in trade is still imitated. This paper applies network science properties and a multi-regional input–output analysis by using the UNCTAD-Eora Global Value Chain Database on the Transport Equipment value added on 2017 to unwrap the specific structural characteristics of the industry. Results show that the industry is highly centralized. The center of the network is dominated by developed countries, mainly from Europe, the United States, and Japan. Emerging countries such as China, Mexico, Thailand, and Poland also have an important position. In addition, the structure reveals two sub-hubs located in East Europe and North America. By extending to community detection, the network consists of three different communities led by Germany, the United States, and the United Kingdom, associated with more significant value-added flows. The study concludes that flows are not always consistent with the economy’s geographical location as usually final goods analysis suggests, and highlight the need to continue using the complex network to reveal the world trade structure.

Emerging in the twenty-first century, Network Science provides practical measures to interpret a system's interactions between the components and their links. Literature has focused on countries' interconnections on the final goods, but its application on the value-added from a network perspective in trade is still imitated. This paper applies network science properties and a multi-regional input–output analysis by using the UNCTAD-Eora Global Value Chain Database on the Transport Equipment value added on 2017 to unwrap the specific structural characteristics of the industry. Results show that the industry is highly centralized. The center of the network is dominated by developed countries, mainly from Europe, the United States, and Japan. Emerging countries such as China, Mexico, Thailand, and Poland also have an important position. In addition, the structure reveals two sub-hubs located in East Europe and North America. By extending to community detection, the network consists of three different communities led by Germany, the United States, and the United Kingdom, associated with more significant value-added flows. The study concludes that flows are not always consistent with the economy's geographical location as usually final goods analysis suggests, and highlight the need to continue using the complex network to reveal the world trade structure. [ABSTRACT FROM AUTHOR]

As a particular logistics service, the express courier service has seen considerable growth recently, which resulted in an unprecedented fierce competition. Besides, the development of information and communication technologies has enabled express company to manage their service. With the purpose of improving service quality and operation efficiency for express company, we focus on the problem of intercity express courier routing in courier-triggered pickup service. A novel pickup policy for courier routing is proposed based on the idea of centrality measures and the nearest-neighbour (NN) policy by considering the un-serviced customer requests as a globally coupled network. This policy enables to dispatch the idle courier to the more central request location, which allows the courier to easily serve the neighbouring requests around the central request location, thus securing both global and local performance. We also propose a simple prototype of real-time fleet management system where the proposed pickup policy is embedded into it. To evaluate the efficiency and practicability of the pickup policy, we conduct comprehensive computational experiments to generate various testing scenarios; moreover, two widely used dispatching policies – NN and first-come-first-served (FCFS) – are considered as the benchmark policy. Results show that the proposed pickup policy significantly outperforms the NN and FCFS policies in terms of waiting time and total service time. [ABSTRACT FROM PUBLISHER]

The network approach represents a novel paradigm for exploring relations between psychological constructs and observable variables. According to this approach, variables form an autonomous dynamical system; the psychological construct is therefore not viewed as their common cause, but a result of their complex interactions. From an analytical point of view, this approach is based on network analysis — a set of procedures which models variables as nodes connected by a set of edges. This paper presents an overview of network analytical procedures. In other words, it offers a brief explanation of the methods, as well as their practical application in two separate datasets. The first dataset represents data on DASS-21 (N = 1016) and it serves to demonstrate network estimation, centrality measures calculation, community detection and network stability analyses. According to the results, the highest centrality was obtained for the depression item I felt that I had nothing to look forward to, anxiety item I felt I was close to panic, and stress item I felt that 1 was using a lot of nervous energy. As expected, nodes were grouped into three clusters, namely Depression, Anxiety and Stress. Stability analyses demonstrated limited stability of edge strength, while the stability of node centrality depended on the measure used. In the second dataset, which represents data on adolescents' attitudes towards one's body appearance, the Network Comparison Test was demonstrated by comparing male (n = 524) and female (n = 763) networks. Results showed that the two networks do not differ substantially.

Abstract Background In the network approach to psychopathology, psychiatric disorders are considered networks of causally active symptoms (nodes), with node centrality hypothesized to reflect symptoms’ causal influence within a network. Accordingly, centrality measures have been used in numerous network-based cross-sectional studies to identify specific treatment targets, based on the assumption that deactivating highly central nodes would proliferate to other nodes in the network, thereby collapsing the network structure and alleviating the overall psychopathology (i.e., the centrality hypothesis). Methods Here, we summarize three types of evidence pertaining to the centrality hypothesis in psychopathology. First, we discuss the validity of the theoretical assumptions underlying the centrality hypothesis in psychopathology. We then summarize the methodological aspects of extant studies using centrality measures as predictors of symptom change following treatment, while delineating their main findings and several of their limitations. Finally, using a specific dataset of 710 treatment-seeking patients with posttraumatic stress disorder (PTSD) as an example, we empirically examine node centrality as a predictor of therapeutic change, replicating the approach taken by previous studies, while addressing some of their limitations. Specifically, we investigated whether three pre-treatment centrality indices (strength, predictability, and expected influence) were significantly correlated with the strength of the association between a symptom’s change and the change in the severity of all other symptoms in the network from pre- to post-treatment (Δnode-Δnetwork association). Using similar analyses, we also examine the predictive validity of two simple non-causal node properties (mean symptom severity and infrequency of symptom endorsement). Results Of the three centrality measures, only expected influence successfully predicted how strongly changes in nodes/symptoms were associated with change in the remainder of the nodes/symptoms. Importantly, when excluding the amnesia node, a well-documented outlier in the phenomenology of PTSD, none of the tested centrality measures predicted symptom change. Conversely, both mean symptom severity and infrequency of symptom endorsement, two standard non-network-derived indices, were found to be more predictive than expected influence and remained significantly predictive also after excluding amnesia from the network analyses. Conclusions The centrality hypothesis in its current form is ill-defined, showing no consistent supporting evidence in the context of cross-sectional, between-subject networks.

When considering complex systems, identifying the most important actors is often of relevance. When the system is modeled as a network, centrality measures are used which assign each node a value due to its position in the network. It is often disregarded that they implicitly assume a network process flowing through a network, and also make assumptions of how the network process flows through the network. A node is then central with respect to this network process (Borgatti in Soc Netw 27(1):55–71, 2005, https://doi.org/10.1016/j.socnet.2004.11.008). It has been shown that real-world processes often do not fulfill these assumptions (Bockholt and Zweig, in Complex networks and their applications VIII, Springer, Cham, 2019, https://doi.org/10.1007/978-3-030-36683-4%5f7). In this work, we systematically investigate the impact of the measures' assumptions by using four datasets of real-world processes. In order to do so, we introduce several variants of the betweenness and closeness centrality which, for each assumption, use either the assumed process model or the behavior of the real-world process. The results are twofold: on the one hand, for all measure variants and almost all datasets, we find that, in general, the standard centrality measures are quite robust against deviations in their process model. On the other hand, we observe a large variation of ranking positions of single nodes, even among the nodes ranked high by the standard measures. This has implications for the interpretability of results of those centrality measures. Since a mismatch of the behaviour of the real network process and the assumed process model does even affect the highly-ranked nodes, resulting rankings need to be interpreted with care. [ABSTRACT FROM AUTHOR]

The purpose of the study is to use a social network analysis of a construction project's participants to identify key participants using centrality measures and identify communities of participants in the network. This article analyses the communication network of a construction project that consists of 34 participants. Analysis of four centrality measures of the network's nodes showed that there was a significant information load for a few key participants. The Eigenvector Centrality was chosen as the most appropriate basic measure of centrality because it takes into account the neighbouring nodes' level of importance. The Louvain clustering method was found to be more effective than the Girvan–Newman method. The Louvain algorithm divided the project communication network into three communities, in which the participants are interconnected by the technological processes and the work performed. A hypothetical example is presented of how the clustering technique can be used to improve project communication. Adding a 'Project Manager Assistant' was selected for addition, and assumptions made to demonstrate how the load might be reduced and effectiveness assessed. These methods of assessing centrality and clustering show potential in project management to analyse a real communication network and when making managerial decisions. [ABSTRACT FROM AUTHOR]

One of the disciplines behind the science of science is the study of scientific networks. This work focuses on scientific networks as a social network having different nodes and connections. Nodes can be represented by authors, articles or journals while connections by citation, co-citation or co-authorship. One of the challenges in creating scientific networks is the lack of publicly available comprehensive data set. It limits the variety of analyses on the same set of nodes of different scientific networks. To supplement such analyses we have worked on publicly available citation metadata from Crossref and OpenCitatons. Using this data a workflow is developed to create scientific networks. Analysis of these networks gives insights into academic research and scholarship. Different techniques of social network analysis have been applied in the literature to study these networks. It includes centrality analysis, community detection, and clustering coefficient. We have used metadata of Scientometrics journal, as a case study, to present our workflow. We did a sample run of the proposed workflow to identify prominent authors using centrality analysis. This work is not a bibliometric study of any field rather it presents replicable Python scripts to perform network analysis. With an increase in the popularity of open access and open metadata, we hypothesise that this workflow shall provide an avenue for understanding scientific scholarship in multiple dimensions.

The network approach represents a novel paradigm for exploring relations between psychological constructs and observable variables. According to this approach, variables form an autonomous dynamical system; the psychological construct is therefore not viewed as their common cause, but a result of their complex interactions. From an analytical point of view, this approach is based on network analysis —a set of procedures which models variables as nodes connected by a set of edges. This paper presents an overview of network analytical procedures. In other words, it offers a brief explanation of the methods, as well as their practical application in two separate datasets. The first dataset represents data on DASS-21 (N = 1016) and it serves to demonstrate network estimation, centrality measures calculation, community detection and network stability analyses. According to the results, the highest centrality was obtained for the depression item I felt that I had nothing to look forward to, anxiety item I felt I was close to panic, and stress item I felt that 1 was using a lot of nervous energy. As expected, nodes were grouped into three clusters, namely Depression, Anxiety and Stress. Stability analyses demonstrated limited stability of edge strength, while the stability of node centrality depended on the measure used. In the second dataset, which represents data on adolescents' attitudes towards one's body appearance, the Network Comparison Test was demonstrated by comparing male (n = 524) and female (n = 763) networks. Results showed that the two networks do not differ substantially. [ABSTRACT FROM AUTHOR]

Since 9/11, terrorism has become a global issue of the twenty-first century. Terrorist organizations become important actors of world politics as they gain influence on political process and decision-making. Some organizations compete with each other in order to gain more power and influence. We study the distribution of power among terrorist groups using network approach and applying classic and new centrality indices (Short-Range (SRIC) and Long-Range interactions indices (LRIC)). These indices allow to identify terrorist groups with direct and indirect influence on the terrorist network. [ABSTRACT FROM AUTHOR]

A basic question in network analysis concerns the quantification of the importance of each node in terms of network connectivity. To this end, a possible approach consists in using cooperative game theory tools to define a measure of node centrality. In this paper, given a transportation network, a cooperative game model with transferable utility (TU game) is considered. The nodes of the network represent the players in such a game, and the Shapley values of the nodes are used to measure centrality. The model, called Transportation Network cooperative (TNc) game, integrates within the utility function of the TU game the network topology and the demand. Properties of TNc games and their associated utility functions are investigated. Numerical results are reported, to get insights into the obtained properties. [ABSTRACT FROM AUTHOR]

Livestock movements are an important mechanism of infectious disease transmission. Where these arewell recorded, network analysis tools have been used to successfully identify system properties, highlight vulnerabilities to transmission, and inform targeted surveillance and control. Here we highlight the main uses of network properties in understanding livestock disease epidemiology and discuss statistical approaches to infer network characteristics from biased or fragmented datasets. We use a 'hurdle model' approach that predicts (i) the probability of movement and (ii) the number of livestock moved to generate synthetic 'complete' networks of movements between administrative wards, exploiting routinely collected government movement permit data from northern Tanzania. We demonstrate that this model captures a significant amount of the observed variation. Combining the cattle movement network with a spatial between-ward contact layer, we create a multiplex, over which we simulated the spread of 'fast' (R0 = 3) and 'slow' (R0 = 1.5) pathogens, and assess the effects of random versus targeted disease control interventions (vaccination andmovement ban). The targeted interventions substantially outperform those randomly implemented for both fast and slow pathogens. Our findings provide motivation to encourage routine collection and centralization of movement data to construct representative networks. [ABSTRACT FROM AUTHOR]

Abstract We first present a comprehensive review of various Markov metrics used in the literature and express them in a consistent framework. We then introduce the fundamental tensor – a generalization of the well-known fundamental matrix – and show that classical Markov metrics can be derived from it in a unified manner. We provide a collection of useful relations for Markov metrics that are useful and insightful for network studies. To demonstrate the usefulness and efficacy of the proposed fundamental tensor in network analysis, we present four important applications: 1) unification of network centrality measures, 2) characterization of (generalized) network articulation points, 3) identification of network's most influential nodes, and 4) fast computation of network reachability after failures. [ABSTRACT FROM AUTHOR]

This article studies about the collaborative measures of published documents in the field of chromosome anomalies. It discusses about inadequacies of collaborative measures in analyzing the collaborating behavior and strength of collaboration in a discipline. It also suggests centrality measures, as degree centrality, closeness, and betweenness in analyzing the collaboration among the researchers and scientists in the field of chromosome anomalies. The bibliographical database PubMed is used as sources for bibliometrics and 35912 citations examined for co - authorship pattern, collaborative behavior of the scientists. Centrality measures were used to construct a network for co - authorship in chromosome anomalies research during the year 2007 - 2016 and to find out the most influential predominant author in the field. [ABSTRACT FROM AUTHOR]

Mountain protected areas (PAs) aim to preserve vulnerable environments and at the same time encourage numerous outdoor leisure activities. Understanding the way people use natural environments is crucial to balance the needs of visitors and site capacities. This study aims to develop an approach to evaluate the structure and use of designated skiing zones in PAs combining Global Positioning System (GPS) tracking and analytical methods based on graph theory. The study is based on empirical data (n = 609 GPS tracks of backcountry skiers) collected in Tatra National Park (TNP), Poland. The physical structure of the entire skiing zones system has been simplified into a graph structure (structural network; undirected graph). In a second step, the actual use of the area by skiers (functional network; directed graph) was analyzed using a graph-theoretic approach. Network coherence (connectivity indices: β, γ, α), movement directions at path segments, and relative importance of network nodes (node centrality measures: degree, betweenness, closeness, and proximity prestige) were calculated. The system of designated backcountry skiing zones was not evenly used by the visitors. Therefore, the calculated parameters differ significantly between the structural and the functional network. In particular, measures related to the actually used trails are of high importance from the management point of view. Information about the most important node locations can be used for planning sign-posts, on-site maps, interpretative boards, or other tourist infrastructure. [ABSTRACT FROM AUTHOR]

Network connectivity is an important aspect of any transportation network, as the role of the network is to provide a society with the ability to easily travel from point to point using various modes. A basic question in network analysis concerns how “important” each node is. An important node might, for example, greatly contribute to short connections between many pairs of nodes, handle a large amount of the traffic, generate relevant information, represent a bridge between two areas, etc. In order to quantify the relative importance of nodes, one possible approach uses the concept of centrality. A limitation of classical centrality measures is the fact that they evaluate nodes based on their individual contributions to the functioning of the network. The present paper introduces a game theory approach, based on cooperative games with transferable utility. Given a transportation network, a game is defined taking into account the network topology, the weights associated with the arcs, and the demand based on an origin-destination matrix (weights associated with nodes). The network nodes represent the players in such a game. The Shapley value, which measures the relative importance of the players in transferable utility games, is used to identify the nodes that have a major role. For several network topologies, a comparison is made with well-known centrality measures. The results show that the suggested centrality measures outperform the classical ones, and provide an innovative approach for transportation network analysis. [ABSTRACT FROM AUTHOR]

In a risk analysis framework, food chain safety measures should be objective and scientifically based. Network science - as a decision support tool - may have an important role in bringing safety to the food supply. The aim of the present work is to develop a network-based assessment methodology for Hungarian cattle holdings. The criteria of which is (1) suitable for risk-based planning in order to put resources into the most critical elements of the cattle production network; (2) should be capable of simulating different epidemiological situations in order to increase preparedness for real epidemics. [ABSTRACT FROM AUTHOR]

Mountain protected areas (PAs) aim to preserve vulnerable environments and at the same time encourage numerous outdoor leisure activities. Understanding the way people use natural environments is crucial to balance the needs of visitors and site capacities. This study aims to develop an approach to evaluate the structure and use of designated skiing zones in PAs combining Global Positioning System (GPS) tracking and analytical methods based on graph theory. The study is based on empirical data (n = 609 GPS tracks of backcountry skiers) collected in Tatra National Park (TNP), Poland. The physical structure of the entire skiing zones system has been simplified into a graph structure (structural network; undirected graph). In a second step, the actual use of the area by skiers (functional network; directed graph) was analyzed using a graph-theoretic approach. Network coherence (connectivity indices: β, γ, α), movement directions at path segments, and relative importance of network nodes (node centrality measures: degree, betweenness, closeness, and proximity prestige) were calculated. The system of designated backcountry skiing zones was not evenly used by the visitors. Therefore, the calculated parameters differ significantly between the structural and the functional network. In particular, measures related to the actually used trails are of high importance from the management point of view. Information about the most important node locations can be used for planning sign-posts, on-site maps, interpretative boards, or other tourist infrastructure.

All over the world, intelligence services are collecting data concerning possible terrorist threats. This information is usually transformed into network structures in which the nodes represent the individuals in the data set and the links possible connections between these individuals. Unfortunately, it is nearly impossible to keep track of all individuals in the resulting complex network. Therefore, Lindelauf et al. (2013) introduced a methodology that ranks terrorists in a network. The rankings that result from this methodology can be used as a decision support system to efficiently allocate the scarce surveillance means of intelligence agencies. Moreover, usage of these rankings can improve the quality of surveillance which can in turn lead to prevention of attacks or destabilization of the networks under surveillance. The methodology introduced by Lindelauf et al. (2013) is based on a game theoretic centrality measure, which is innovative in the sense that it takes into account not only the structure of the network but also individual and coalitional characteristics of the members of the network. In this paper we elaborate on this methodology by introducing a new game theoretic centrality measure that better takes into account the operational strength of connected subnetworks. Moreover, we perform a sensitivity analysis on the rankings derived from this new centrality measure for the case of Al Qaeda's 9/11 attack. In this sensitivity analysis we consider firstly the possible additional information available about members of the network, secondly, variations in relational strength and, finally, the absence or presence of a small percentage of links in the network. We also introduce a case specific method to compare the different rankings that result from the sensitivity analysis and show that the new centrality measure is robust to small changes in the data. [ABSTRACT FROM AUTHOR]

Students' understanding of scientific conceptual knowledge is often represented as an interlinked web of concepts, principles, laws and models. A long-standing problem in educational research is identifying the key concepts that are central in producing cohesion and contingency in such a web. Here we use network analysis to examine students' representations of the relatedness of physics concepts in the form of concept maps, and suggest how key concepts can be identified. The concept maps are analysed as weighted networks, where nodes are concepts or other conceptual elements and links represent different types of epistemically justified connections between concepts. The importance of concepts in providing cohesion is operationalized through subgraph centrality ${\mathrm {SC}}$, while their importance in providing contingency is operationalized through communicability betweenness centrality ${\mathrm {BC}}$. Key concepts are identified through importance ranking ${\mathrm {IR}}$, which is the geometric mean of ${\mathrm {SC}}$ and ${\mathrm {BC}}$, suitably normalized. We show that ${\mathrm {IR}}$ is able to reliably identify a set of nodes that are the most important in all networks. In order to effect a more detailed comparison of different concept networks, a similarity measure is developed, which pays attention to subtle but important differences in the importance rankings of concepts in different concept networks. [ABSTRACT FROM AUTHOR]