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We describe a methodology for characterizing the relative structural importance of an arbitrary network edge by exploiting the properties of a k-shortest path algorithm. We introduce the metric Edge Gravity, measuring how often an edge occurs in any possible network path, as well as k-Gravity, a lower bound based on paths enumerated while solving t...
Citations
... This can be achieved by representing interactions between market participants in the form of networks of the transactions they undertake [1,2]. Important nodes can then be identified by considering concepts such as 'centrality', for which there are a number of measures to rank nodes according to their position in the network [3,4]. ...
... We observed fairly high levels of class imbalance across the three datasets -Equity-1 showing 1499 of 2063 present in the subsequent snapshot, Equity-2 showed 724 of 880 present, and Equity-3 showed 1803 of 2237 present, so we applied a random over-sampling strategy to correct this for all three datasets. We made use of 5-fold cross validation 3 to select the best classifier and its associated parameter values 4 . This not only allows us to assess whether our measures of node importance are predictive of subsequent node activity, but it also provides us with the means of comparing the different entries of the feature vector through their feature importances. ...
... For this, we split our data into training and test sets, with a 40-40-20 train, validation, test split. We split the data whilst keeping the ordering of time, so that the model is not trained on data from the future.4 Both logistic regression and random forest classifiers were considered to allow for potentially non-linear relationships. ...
A fundamental problem in the study of networks is the identification of important nodes. This is typically achieved using centrality metrics, which rank nodes in terms of their position in the network. This approach works well for static networks, that do not change over time, but does not consider the dynamics of the network. Here we propose instead to measure the importance of a node based on how much a change to its strength will impact the global structure of the network, which we measure in terms of the spectrum of its adjacency matrix. We apply our method to the identification of important nodes in equity transaction networks, and we show that, while it can still be computed from a static network, our measure is a good predictor of nodes subsequently transacting. This implies that static representations of temporal networks can contain information about their dynamics.
... Although the bulk of the attention has focused on importance of actors in networks, Helander et al. [18] propose a method for characterising the relative importance of an edge, which they refer to as edge gravity. Edge gravity measures how often an edge occurs in any possible network path. ...
... These observations indicate that the static structural importance can be indicative of the presence of a subsequent change, however more work is needed to understand the shape of the distribution and the identification of different l e regimes. We will also consider taking a similar approach with other measures of edge importance, for example edge gravity [18]. More work is also needed to understand the subsequent impact on the global network structure of an edge changing. ...
To monitor risk in temporal financial networks, we need to understand how individual behaviours affect the global evolution of networks. Here we define a structural importance metric—which we denote as $l_{e}$ l e —for the edges of a network. The metric is based on perturbing the adjacency matrix and observing the resultant change in its largest eigenvalues. We then propose a model of network evolution where this metric controls the probabilities of subsequent edge changes. We show using synthetic data how the parameters of the model are related to the capability of predicting whether an edge will change from its value of $l_{e}$ l e . We then estimate the model parameters associated with five real financial and social networks, and we study their predictability. These methods have applications in financial regulation whereby it is important to understand how individual changes to financial networks will impact their global behaviour. It also provides fundamental insights into spectral predictability in networks, and it demonstrates how spectral perturbations can be a useful tool in understanding the interplay between micro and macro features of networks.
... Although the bulk of the attention has focused on importance of actors in networks, Helander et. al. [17] propose a method for characterising the relative importance of an edge, which they refer to as edge gravity. Edge gravity measures how often an edge occurs in any possible network path. ...
... These observations indicate that the static structural importance can be indicative of the presence of a subsequent change, however more work is needed to understand the shape of the distribution and the identification of different l e regimes. We will also consider taking a similar approach with other measures of edge importance, for example edge gravity [17]. More work is also needed to understand the subsequent impact on the global network structure of an edge changing. ...
To monitor risk in temporal financial networks, we need to understand how individual behaviours affect the global evolution of networks. Here we define a structural importance metric - which we denote as $l_e$ - for the edges of a network. The metric is based on perturbing the adjacency matrix and observing the resultant change in its largest eigenvalues. We then propose a model of network evolution where this metric controls the probabilities of subsequent edge changes. We show using synthetic data how the parameters of the model are related to the capability of predicting whether an edge will change from its value of $l_e$. We then estimate the model parameters associated with five real financial and social networks, and we study their predictability. These methods have application in financial regulation whereby it is important to understand how individual changes to financial networks will impact their global behaviour. It also provides fundamental insights into spectral predictability in networks, and it demonstrates how spectral perturbations can be a useful tool in understanding the interplay between micro and macro features of networks.
... Social network researchers and theorists have studied edge strengths/weaknesses and their roles ever since Granovetter [25] first described how a job candidate's weakest ties have utility in a job search [12,[26][27][28][29][30][31][32][33][34]. Definitions have been offered for four hierarchical edge strength types in an identification algorithm context: bond edges, kth-layer local bridges, global bridges, and silk edges [35]. ...
Many real-world networks consisting of nodes representing (in)tangible asymmetric information or energy flows must be modeled as directed graphs (digraphs). Several methods for classifying non-directional edges in terms of strong or weak ties have been developed for well-known non-directional networks, but none specifically for directed networks. In almost all cases, definitions and identification methods are simple, incomplete, reliant on intuition, and based on the assumption that anything that is not weak must be strong. Researchers have generally failed to consider overlapping and hierarchical community properties that accurately reflect organizational structures or the functional components commonly found in real-world complex networks, resulting in multiple challenges to analyzing many types of directed networks. In this paper we describe a method that considers asymmetric definitions of arc strength, especially when arcs hold important directional significance. To more fully capture overlapping and hierarchical network community structures, we used hierarchy-based definitions to identify bond arcs, kth-layer local bridges, global bridges, and silk arcs and to create a hierarchical arc type analysis (HATA) algorithm. The algorithm employs a mix of common middle node measures and statistical parameters generated by randomized directed networks corresponding to the network being investigated. To test the HATA algorithm, we conducted four experiments involving a mix of arc rewiring and additions, multiple datasets associated with the Travian game, 56 empirical networks from previous studies, and 3 bird song transition networks. Our results indicate that HATA offers a novel perspective to understanding arc strengths and structures in directed complex networks.
... Whereas at the macro level, they comprise a directed network of flows of goods from resources to consumption, at the micro level they represent an undirected network of communities of practices among goods. Given that "the domain of change in an evolutionary process is neither micro nor macro but meso" (Dopfer et al. 2004: 271), calculating the relative relevance of each node (through "centrality-measures", see Jackson 2008) and each edge (through "edge gravity", see Helander and McAllister 2018) for the overall network architecture at the meso-level promises to identify leverage points for possible transformation of the economic system of interest. [A]ll boundaries are arbitrary." ...
... Their overall scores furthermore allow distinguishing the relevance of key products relative to each, as indicated by the size of nodes in fig. 7. The 38 bio-based key products are connected by 182 different markets (edges) in total. Forming an undirected network through mutual exchange in the market by communities of practices (see fig. 4), the relevance of each of those markets (edges) for the overall network architecture can be determined through their edge gravity measure (Helander and McAllister 2018). ...
... Complementing classical SNA metrics such as edge betweenness and edge gravity quantifies the importance of an edge for the overall network architecture independent from the importance of the nodes, "[…] based on how often that edge appears in any path, rather than restricting attention to only the shortest paths, as in the case of edge betweenness" (Helander and McAllister 2018: 3). To calculate the edge gravity, an adaptation of the "k-shortest path algorithm" is used that allows the exact computation of the edge gravity metric in polynomial time (Helander and McAllister 2018). ...
This paper discusses a method for the identification of leverage points for the transformation of complex economic systems by combining value chain analysis and a relational perspective on markets. We introduce the Network of Markets Approach and show that it is central in bridging the two concepts and apply social network analysis in order to do so. The case of the German bioeconomy, the transition from a fossil to a bio-based economy, is chosen to exemplify the application of the method. Here, we provide the specific rationales behind our approach and identify possible key markets for governance interventions in the transformation toward a bio-based economy in Germany.
A fundamental problem in the study of networks is the identification of important nodes. This is typically achieved using centrality metrics, which rank nodes in terms of their position in the network. This approach works well for static networks, that do not change over time, but does not consider the dynamics of the network. Here we propose instead to measure the importance of a node based on how much a change to its strength will impact the global structure of the network, which we measure in terms of the spectrum of its adjacency matrix. We apply our method to the identification of important nodes in equity transaction networks and show that, while it can still be computed from a static network, our measure is a good predictor of nodes subsequently transacting. This implies that static representations of temporal networks can contain information about their dynamics.
The challenge for a circular plastics economy transition is to focus policies on key leverage points that initiate actual system transitions. This requires a systemic perspective on the plastics industries. This study takes such a systemic perspective by employing a network approach to examine the often-underestimated complexity of interrelating markets in a circular plastics economy, and their structural sensitivity to governance interventions. Based on the case of polyethylene terephthalate (PET) markets in Germany, we investigate the structures and underlying dynamics of increasing circularity in the PET industry. Concerns about plastic litter accumulating in the natural environment have facilitated the development of niche markets for the recycling of plastic litter recovered from the environment. We systematically reveal that recycling markets connecting diverse waste sources with a broad range of new applications are key areas of intervention in the structural transitions towards circular industries. By connecting otherwise disconnected parts of the system, the recycling of recovered plastic litter is a key leverage point for the circular economy transition. We recommend to focus governance efforts on such key leverage markets as powerful venues to initiate systemic change.
