Average Shortest Path Length Research Articles

Dynamical network-based evaluation for neuromuscular dysfunction in stroke-induced hemiplegia during standing

BackgroundA given movement requires precise coordination of multiple muscles under the control of center nervous system. However, detailed knowledge about the changing characteristics of neuromuscular control for multi-muscle coordination in post-stroke hemiplegic patients during standing is still lacking. This study aimed to investigate the hemiplegia-linked neuromuscular dysfunction during standing from the perspective of multi-muscle dynamical coordination by utilizing a novel network approach – weighted recurrence network (WRN).MethodsTen male hemiplegic patients with first-ever stroke and 10 age-matched healthy male adults were instructed to stand on a platform quietly for 30 s with eyes opened and eyes closed, respectively. The WRN was constructed based on the surface electromyography signals of 16 muscles from trunk, hips, thighs and calves. Relevant topological parameters, including clustering coefficient (C) and average shortest path length (L), were extracted to evaluate the dynamical coordination of multiple muscles. A measure of node centrality in network theory, degree of centrality (DC), was innovatively introduced to assess the contribution of single muscle in the multi-muscle dynamical coordination. The standing-related assessment metric, center of pressure (COP), was provided by the platform directly.ResultsResults showed that the post-stroke hemiplegic patients stood with remarkably higher similarity of muscle activation and more coupled intermuscular dynamics, characterized by higher C and lower L than the healthy subjects (p < 0.05). The DC values and rankings of back, hip and calf muscles on the affected side were significantly decreased, whereas those on the unaffected side were significantly increased in hemiplegia group compared with the healthy group (p < 0.05). Without visual feedback, subjects exhibited enhanced muscle coordination and increased muscle involvement (p < 0.05). A decrease in C and an increase in L of WRN were observed with decreased COP areas (p < 0.05).ConclusionsThese findings revealed that stroke-induced hemiplegia could significantly influence the neuromuscular control, which was manifested as more coupled intermuscular dynamics, abnormal deactivation of muscles on affected side and compensation of muscles on unaffected side from the perspective of multi-muscle coordination. Enhanced multi-muscle dynamical coordination was strongly associated with impaired postural control. This study provides a novel analytical tool for evaluation of neuromuscular dysfunction and specification of responsible muscles for impaired postural control in stroke-induced hemiplegic patients, and could be potentially applied in clinical practice.

Dynamics of second-language learners' semantic memory networks: Evidence from a snowball sampling paradigm.

This article provides an analysis of structural changes in second-language (L2)-based semantic memory networks-graphs composed of L2 words as nodes and semantic relations between them as edges, during L2 learning. We used snowball sampling paradigm to create individual semantic networks of participants divided into two groups differing in L2 learning time and then compare their structural characteristics cross-sectionally. The results showed that as L2 learning progresses, semantic memory networks tend to become more connected (by increasing the average node degree), more efficient (by decreasing the average shortest path length), less fragmented (by decreasing the modularity), less centralized (by decreasing the centralization), less dense (by decreasing the density), and no more "small-worlded" (by similar average clustering coefficients and small-world indices). The findings provide quantitative evidence of how the duration of L2 learning shapes the structure of L2-based semantic memory networks generated in the snowball sampling paradigm. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

RGP: Neural Network Pruning Through Regular Graph With Edges Swapping.

Deep learning technology has found a promising application in lightweight model design, for which pruning is an effective means of achieving a large reduction in both model parameters and float points operations (FLOPs). The existing neural network pruning methods mostly start from the consideration of the importance of model parameters and design parameter evaluation metrics to perform parameter pruning iteratively. These methods were not studied from the perspective of network model topology, so they might be effective but not efficient, and they require completely different pruning for different datasets. In this article, we study the graph structure of the neural network and propose a regular graph pruning (RGP) method to perform a one-shot neural network pruning. Specifically, we first generate a regular graph and set its node-degree values to meet the preset pruning ratio. Then, we reduce the average shortest path-length (ASPL) of the graph by swapping edges to obtain the optimal edge distribution. Finally, we map the obtained graph to a neural network structure to realize pruning. Our experiments demonstrate that the ASPL of the graph is negatively correlated with the classification accuracy of the neural network and that RGP has a strong precision retention capability with high parameter reduction (more than 90%) and FLOPs reduction (more than 90%) (the code for quick use and reproduction is available at https://github.com/Holidays1999/Neural-Network-Pruning-through-its-RegularGraph-Structure).

Structure and dynamics of growing networks of Reddit threads

Millions of people use online social networks to reinforce their sense of belonging, for example by giving and asking for feedback as a form of social validation and self-recognition. It is common to observe disagreement among people beliefs and points of view when expressing this feedback. Modeling and analyzing such interactions is crucial to understand social phenomena that happen when people face different opinions while expressing and discussing their values. In this work, we study a Reddit community in which people participate to judge or be judged with respect to some behavior, as it represents a valuable source to study how users express judgments online. We model threads of this community as complex networks of user interactions growing in time, and we analyze the evolution of their structural properties. We show that the evolution of Reddit networks differ from other real social networks, despite falling in the same category. This happens because their global clustering coefficient is extremely small and the average shortest path length increases over time. Such properties reveal how users discuss in threads, i.e. with mostly one other user and often by a single message. We strengthen such result by analyzing the role that disagreement and reciprocity play in such conversations. We also show that Reddit thread’s evolution over time is governed by two subgraphs growing at different speeds. We discover that, in the studied community, the difference of such speed is higher than in other communities because of the user guidelines enforcing specific user interactions. Finally, we interpret the obtained results on user behavior drawing back to Social Judgment Theory.

Peanut–cotton intercropping to enhance soil ecosystem multifunctionality: Roles of microbial keystone taxa, assembly processes, and C-cycling profiles

Legume-based intercropping, such as the peanut–cotton system, stands out as a promising strategy for enhancing soil ecosystem multifunctionality (EMF); however, the underlying microbial mechanisms driving these enhancements remain inadequately explored. In this study, after implementing peanut–cotton intercropping for six consecutive years, a data set of 13 ecosystem functional indicators including 41 soil variables, was obtained and used to quantify the average EMF index. We investigated changes in microbial keystone taxa in co-occurrence networks, community assembly processes, carbon (C) cycling profiles, and their collective impacts on soil EMF. Soil EMF increased by an average of 140.0 % in the peanut–cotton intercropping system, compared with monoculture systems of both peanut and cotton, driven by significant increases in C-cycling (159.9 %), nutrient provisioning (91.2 %), and microbial growth efficiency functions (53.9 %). The peanut–cotton intercropping system significantly increased the average well-color developments (AWCD), abundance of C-fixation and C-degradation genes, and related pathways, resulting in a highly vigorous microbial C-cycling profile. The microbial community assembly processes shifted from a balance of stochastic and deterministic processes in monocultures to predominantly deterministic processes (> 70 %) in the intercropping system. Additionally, the peanut–cotton intercropping system fostered a more efficient and stable bacterial–fungal cross-kingdom network than the monocultures, characterized by a higher average clustering coefficient, higher robustness, and shorter average path length. This intercropping system also recruited a group of keystone taxa affiliated with Proteobacteria, Actinobacteria, and Ascomycota phyla. The enhancement of EMF in the peanut–cotton intercropping system resulted from the positive impact of key microbial community members and their assembly, C/N ratios, AWCD, and C-fixation and C-degradation genes. Our study provides insights into the complex ecological linkages between microbial communities, C-cycling profiles, and soil ecosystem functions, providing valuable insights into the microbial mechanisms underlying the benefits of intercropping systems.

Exposure pattern of triclosan and tetracycline change their impacts on methanogenic digestion microbiomes

Triclosan (TCS) and tetracycline (TC) as common antibacterial agents are frequently detected in the influent of wastewater treatment plants. The TCS and TC exposure patterns may determine their impacts on wastewater treatment microbiomes, on which information remains unknown. In this study, the impacts of sequential exposure of TCS and TC on methanogenic digestion microbiomes in upflow anaerobic sludge blanket (UASB) reactors were analyzed and compared with that of the same microbiomes being simultaneously exposed to TCS and TC. Results indicated that the UASB reactor 2 (MD2) with sequential TCS-TC exposure consistently demonstrated higher chemical oxygen demand (COD) removal efficiency (94.7 %). In contrast, in the MD1 reactor, COD removal efficiency decreased from 94.4 % to 82.7 % upon simultaneous exposure to TCS and TC. Accordingly, a 1.8 times higher enrichment of total antibiotic resistance genes (ARGs) was observed in MD1 relative to MD2. Using a dissimilarity-overlap approach, the MD2 microbiome with sequential exposure was predominantly mediated by deterministic factors in their community assembly (largely contributed by abundant and intermediate biospheres), resulting in microbial interaction networks with higher average clustering coefficients and shorter average path lengths, compared to the MD1 microbiomes. Our results could support sustainable management of TCS and TC contamination in wastewater treatment plants.

Neuromuscular conditions in post-stroke ankle-foot dysfunction reflected by surface electromyography

BackgroundRating scales and linear indices of surface electromyography (sEMG) cannot quantify all neuromuscular conditions associated with ankle-foot dysfunction in hemiplegic patients. This study aimed to reveal potential neuromuscular conditions of ankle-foot dysfunction in hemiplegic patients by nonlinear network indices of sEMG.MethodsFourteen male patients with hemiplegia and 10 age- and sex-matched healthy male adults were recruited and tested in static standing position. The characteristics of the root mean square (RMS), median frequency (MF), and three nonlinear indices, the clustering coefficient (C), the average shortest path length (L), and the degree centrality (DC), of eight groups of muscles in bilateral calves were observed.ResultsCompared to those of the control group, the RMS of the medial gastrocnemius (MG), flexor digitorum longus (FDL), and extensor digitorum longus (EDL) on the affected side were significantly lower (P < 0.05), and the RMS of the tibial anterior (TA) and EDL on the unaffected side were significantly higher (P < 0.05). The MF of the EDL on the affected side was significantly higher than that on the control side (P < 0.05). The C of the unaffected side was significantly higher than that of the control group, whereas the L was lower (P < 0.05). Compared to those of the control group, the DC of the TA, EDL, and soleus (SOL) on the unaffected sides were higher (P < 0.05), and the DC of the MG on the affected sides was lower (P < 0.05).ConclusionThe change trends and clinical significance of these three network indices, including C, L, and DC, are not in line with those of the traditional linear indices, the RMS and the MF. The C and L may reflect the degree of synchronous activation of muscles during a certain motor task. The DC might be able to quantitatively assess the degree of muscle involvement and reflect the degree of involvement of a single muscle. Linear and nonlinear indices may reveal more neuromuscular conditions in hemiplegic ankle-foot dysfunction from different aspects.Trial registrationChiCTR2100055090.

Node and edge centrality based failures in multi-layer complex networks

Multi-layer complex networks (MLCN) appears in various domains, such as, transportation, supply chains, etc. Failures in MLCN can lead to major disruptions in systems. Several research have focussed on different kinds of failures, such as, cascades, their reasons and ways to avoid them. This paper considers failures in a specific type of MLCN where the lower layer provides services to the higher layer without cross layer interaction, typical of a computer network. A three layer MLCN is constructed with the same set of nodes where each layer has different characteristics, the bottom most layer is Erdos–Renyi (ER) random graph with shortest path hop count among the nodes as gaussian, the middle layer is ER graph with higher number of edges from the previous, and the top most layer is preferential attachment graph with even higher number of edges. Both edge and node failures are considered. Failures happen with decreasing order of centralities of edges and nodes in static batch mode and when the centralities change dynamically with progressive failures. Emergent pattern of three key parameters, namely, average shortest path length (ASPL), total shortest path count (TSPC) and total number of edges (TNE) for all the three layers after node or edge failures are studied. Extensive simulations show that all but one parameters show definite degrading patterns. Surprising, ASPL for the middle layer starts showing a chaotic behaviour beyond a certain point for all types of failures.

Characterising the low-cost carrier network in Southeast Asia: a complex network analysis

This study analyses the low-cost carrier network (LCCN) in Southeast Asia using complex network analysis. An analysis is conducted using the data from 2010 to 2018 covering 442 airports connected by 2,296 air routes. The LCCN is a scale-free network with a degree distribution following a power law in which fewer airports have high connectivity. It is a sparse network with a low network density. The network is efficient with a small average shortest path length, but low intra-connectivity amongst airports in the same cluster. This suggests that with good affordability and low costs, low-cost carriers play an important role in providing essential air connectivity to small and distanced destinations. Although low-cost carriers tend to use a point-to-point network instead of a hub-and-spoke network, major airports still play essential roles in shaping network connectivity and performance. Therefore, airports must have ample capacity and good infrastructure to ensure the network’s performance.

How retrieval processes change with age: Exploring age differences in semantic network and retrieval dynamics.

This study investigated the impact of age on semantic memory networks and retrieval dynamics using a single-list free recall paradigm, involving 318 participants. The younger group, with 175 participants aged 25-55 years (M = 46.68 years; SD = 10.69), and the older group, consisting of 143 participants aged 61-88 years (M = 68.71 years; SD = 6.09), completed a word recall test to assess delayed recall performance. Semantic memory networks were constructed from recall data by analyzing the co-occurrence and sequence of recalled words. We observed significant differences in network structure, where the older group displayed higher average shortest path length and modularity values, indicative of less integrated networks, while the younger group exhibited a higher clustering coefficient, suggesting a more interconnected network. In terms of retrieval dynamics, both groups showed a temporal contiguity effect with forward asymmetry. However, this effect was less pronounced in older adults. The study also identified participants that diverted from the average dynamic curves: one subgroup relied on nontemporal mechanisms, and the other employed a backward direction in memory search. Participants utilizing forward temporal associations demonstrated the highest recall performance. Overall, our findings suggest that lower free recall performance in older adults may be related to a diminished capacity to reinstate temporal context for retrieval and distinct differences in their semantic memory network structure. Specifically, older adults appear to exhibit networks with a less flexible, small-world-like structure. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Predicting herb-disease associations using network-based measures in human protein interactome

BackgroundNatural herbs are frequently used to treat diseases or to relieve symptoms in many countries. Moreover, as their safety has been proven for a long time, they are considered as main sources of new drug development. However, in many cases, the herbs are still prescribed relying on ancient records and/or traditional practices without scientific evidences. More importantly, the medicinal efficacy of the herbs has to be evaluated in the perspective of MCMT (multi-compound multi-target) effects, but most efforts focus on identifying and analyzing a single compound experimentally. To overcome these hurdles, computational approaches which are based on the scientific evidences and are able to handle the MCMT effects are needed to predict the herb-disease associations.ResultsIn this study, we proposed a network-based in silico method to predict the herb-disease associations. To this end, we devised a new network-based measure, WACP (weighted average closest path length), which not only quantifies proximity between herb-related genes and disease-related genes but also considers compound compositions of each herb. As a result, we confirmed that our method successfully predicts the herb-disease associations in the human protein interactome (AUROC = 0.777). In addition, we observed that our method is superior than the other simple network-based proximity measures (e.g. average shortest and closest path length). Additionally, we analyzed the associations between Brassica oleracea var. italica and its known associated diseases more specifically as case studies. Finally, based on the prediction results of the WACP, we suggested novel herb-disease pairs which are expected to have potential relations and their literature evidences.ConclusionsThis method could be a promising solution to modernize the use of the natural herbs by providing the scientific evidences about the molecular associations between the herb-related genes targeted by multiple compounds and the disease-related genes in the human protein interactome.

Power law of path multiplicity in complex networks.

Complex networks describe a wide range of systems in nature and society. As a fundamental concept of graph theory, the path connecting nodes and edges plays a vital role in network science. Rather than focusing on the path length or path centrality, here we draw attention to the path multiplicity related to decision-making efficiency, which is defined as the number of shortest paths between node pairs and thus characterizes the routing choice diversity. Notably, through extensive empirical investigations from this new perspective, we surprisingly observe a "hesitant-world" feature along with the "small-world" feature and find a universal power-law of the path multiplicity, meaning that a small number of node pairs possess high path multiplicity. We demonstrate that the power-law of path multiplicity is much stronger than the power-law of node degree, which is known as the scale-free property. Then, we show that these phenomena cannot be captured by existing classical network models. Furthermore, we explore the relationship between the path multiplicity and existing typical network metrics, such as average shortest path length, clustering coefficient, assortativity coefficient, and node centralities. We demonstrate that the path multiplicity is a distinctive network metric. These results expand our knowledge of network structure and provide a novel viewpoint for network design and optimization with significant potential applications in biological, social, and man-made networks.

A Dynamic Evolutionary Analysis of the Vulnerability of Global Food Trade Networks

The global food trade network (FTN) is a critical infrastructure for achieving the Sustainable Development Goals (SDGs). The FTN’s vulnerability to geopolitical conflicts, public health crises, and climate change events directly impacts food security and the ability to meet the SDGs. This study aims to analyze the dynamic evolution of the vulnerability of FTN, focusing on the period from 2000 to 2022, to aim for strategies for enhancing the resilience and sustainability of the global food system. Based on complex network analysis, we examine the structural characteristics and evolution of FTN for four major crops: soybeans, wheat, rice, and maize. We identify a trend towards increased network density and regionalization, with a decline in average shortest path length (ASPL) and an increase in the average clustering coefficient (ACC). These changes indicate a shift towards a more interconnected and resilient FTN in response to various shocks, including the COVID-19 pandemic and the Russia–Ukraine conflict. The findings suggest that the global FTN has adapted to increase resilience, which is essential for achieving the SDGs related to food security and sustainable development. The study’s insights can guide policy interventions to further strengthen the network against future shocks and promote global food security.

On the transient and equilibrium features of growing fractal complex networks

Complex networks have certain properties that distinguish them from their respective uniform or regular counterparts. One of these properties is the variation of topological properties along different hierarchical levels. In this work, we study how networks that are constructed by repeatedly incorporating a given motif exhibit this property. A motif is henceforth understood as a small subgraph with a reference node where the incorporation respectively occurs. We generate fractal networks using different motifs and observe how their structures change along the growth stages. Two regimes are respectively identified: transient and equilibrium. The former is characterized by significant topological changes that depend on the motif topology, while the equilibrium regime shows more stable, parallel trajectories. A more systematic analysis revealed that the betweenness centrality and the average shortest path lengths were the main topological properties accounting for the changes along the network growth.

A New Approach to Computationally- Successful Linear and Polynomial Regression Analytics of Large Data in Medicine

Features like assortative mixing; high clustering, short average path lengths, broad degree distributions, and community structure have been the subject of numerous recent social network studies. All of these features are met by the model that is introduced in this study. Additionally, our model enables interactions between various communities, fostering a rich network environment. The asymptotically scale-free degree distribution is maintained by our model, which achieves a high clustering coefficient. In our model, the community structure is generated by a mix of mechanisms involving implicit preferential attachment and random attachment. We expand our consideration to include neighbor of neighbor of Initial Contact (NNIC) as well, in contrast to earlier approaches that solely focused on neighbor of Initial Contact (NIC) as an implicit preferential contact. If a newly added vertex chooses more than one initial contact, this extension makes it possible for contacts between those initial contacts to occur. Consequently, our model facilitates the development of complex social networks beyond those used as basic references. Finally, we conduct centrality calculations on both the existing model and our developed model, providing a comparative analysis of the results.

A Novel Approach for Analyzing Temporal Influence Dynamics in Social Networks

Features like assortative mixing; high clustering, short average path lengths, broad degree distributions, and community structure have been the subject of numerous recent social network studies. All of these features are met by the model that is introduced in this study. Additionally, our model enables interactions between various communities, fostering a rich network environment. The asymptotically scale-free degree distribution is maintained by our model, which achieves a high clustering coefficient. In our model, the community structure is generated by a mix of mechanisms involving implicit preferential attachment and random attachment. We expand our consideration to include neighbor of neighbor of Initial Contact (NNIC) as well, in contrast to earlier approaches that solely focused on neighbor of Initial Contact (NIC) as an implicit preferential contact. If a newly added vertex chooses more than one initial contact, this extension makes it possible for contacts between those initial contacts to occur. Consequently, our model facilitates the development of complex social networks beyond those used as basic references. Finally, we conduct centrality calculations on both the existing model and our developed model, providing a comparative analysis of the results.

Analysis of the Network Efficiency of Chinese Ports in Global Shipping under the Impacts of Typhoons

With the increasing volume of international trade and maritime demand, the requirements for the stability and reliability of the global shipping system are also increasing. The research on the network efficiency of Chinese ports for global shipping can not only examine the importance of Chinese ports in the shipping network but also find out the aspects that need to be improved in the construction of the port’s climate adaptability in the resilience assessment to strengthen port construction and further improve the efficiency of the network. The current study builds a shipping network based on RCEP and systematically examines the key ports in China within the networks. The research paper aims to improve the resilience of the ports and the whole shipping network in response to typhoon disasters. As such, this paper focuses on shipping research based on complex networks and network multi-centricity analysis, followed by a ranking of ports. Firstly, this paper uses UCINET 6 software to build a global shipping network. Such a network evaluates the centrality of ports, calculates the degree of centrality, proximity to centrality, and centrality, and scores them according to the ranking. Then, it selects the top 20 ports in China according to the ranking and researches network efficiency for the listed ports considering the typhoon risks. The analysis of network robustness, average shortest path length, and network efficiency are carried out for the shipping network and China’s essential port nodes in the network. According to the experimental results, no matter the robustness, average shortest path length, or network efficiency, when the important ports of China in the shipping network are affected, they will cause different degrees of impact, and the performance loss caused by multiple ports is higher than that of a single port. They emphasise the significant impact of typhoons on multiple ports and remind people to minimise losses as much as possible based on experimental results, ensuring the stable operation of ports and improving resilience in typhoon prevention under the changing climate. Additionally, they provide a solid foundation to further sustain global shipping network resilience.

A network model of social contacts with small-world and scale-free features, tunable connectivity, and geographic restrictions.

Small-world networks and scale-free networks are well-known theoretical models within the realm of complex graphs. These models exhibit "low" average shortest-path length; however, key distinctions are observed in their degree distributions and average clustering coefficients: in small-world networks, the degree distribution is bell-shaped and the clustering is "high"; in scale-free networks, the degree distribution follows a power law and the clustering is "low". Here, a model for generating scale-free graphs with "high" clustering is numerically explored, since these features are concurrently identified in networks representing social interactions. In this model, the values of average degree and exponent of the power-law degree distribution are both adjustable, and spatial limitations in the creation of links are taken into account. Several topological metrics are calculated and compared for computer-generated graphs. Unexpectedly, the numerical experiments show that, by varying the model parameters, a transition from a power-law to a bell-shaped degree distribution can occur. Also, in these graphs, the degree distribution is most accurately characterized by a pure power-law for values of the exponent typically found in real-world networks.

Multiple sclerosis is associated with differences in semantic memory structure.

Although language is often considered to be largely intact in multiple sclerosis (MS), word-finding difficulties are a common complaint. Recent work suggests that declines in language are not solely the result of motoric and cognitive slowing that is most strongly associated with MS. Network science approaches have been effectively used to examine network structure as it relates to clinical conditions, aging, and language. The present study utilizes a network science approach to investigate whether individuals with MS exhibit less interconnected and resilient semantic networks compared to age-matched neurotypical peers. We used semantic fluency data from 89 participants with MS and 88 neurotypical participants to estimate and analyze the semantic network structure for each participant group. Additionally, we conducted a percolation analysis to examine the resilience of each network. Network measures showed that individuals with MS had lower local and global clustering coefficients, longer average shortest path lengths, and higher modularity values compared to neurotypical peers. Small-worldness, network portrait divergence measures, and community detection analyses were consistent with these results and indicated that macroscopic properties of the two networks differed and that the semantic network for individuals with MS was more fractured than the neurotypical peer network. Moreover, a spreading activation simulation and percolation analysis suggested that the semantic networks of individuals with MS are less flexible and activation degrades faster than those of age-matched neurotypical participants. These differing semantic network structures suggest that language retrieval difficulties in MS partially result from decline in language-specific factors. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Effect of network size on comparing different stock networks

We analyzed complex networks generated by the threshold method in the Korean and Indian stock markets during the non-crisis period of 2004 and the crisis period of 2008, while varying the size of the system. To create the stock network, we randomly selected N stock indices from the market and constructed the network based on cross-correlation among the time series of stock prices. We computed the average shortest path length L and average clustering coefficient C for several ensembles of generated stock networks and found that both metrics are influenced by network size. Since L and C are affected by network size N, a direct comparison of graph measures between stock networks with different numbers of nodes could lead to erroneous conclusions. However, we observed that the dependency of network measures on N is significantly reduced when comparing larger networks with normalized shortest path lengths. Additionally, we discovered that the effect of network size on network measures during the crisis period is almost negligible compared to the non-crisis periods.