No-feedback Condition Research Articles

Obsessive-compulsive tendencies predict seeking proxies for understanding

Background and objectivesThe Seeking Proxies for Internal States (SPIS) model of obsessive-compulsive disorder (OCD) postulates that obsessive-compulsive (OC) individuals have reduced access to their internal states and must therefore seek and rely on external proxies for these states. The present study extended this hypothesis to the feeling of understanding, which had not been examined previously in relation to OCD. MethodsWe presented 148 participants with a computerized task requiring them to read and understand a text on medieval architecture. Participants were randomly assigned to an ongoing feedback condition (comprehension quiz and answers provided after each text segment) or no-feedback condition (quiz and answers provided only at the end). Throughout, participants were offered proxies in the form of “learning aids,” which were unrelated to text comprehension. Participants were divided to high vs. low OC tendencies based on a median split on a measure of OCD symptoms. ResultsAs predicted, lacking feedback on understanding was associated with higher use of proxies and high OC participants used more proxies than low OC participants in the no-feedback condition. Actual understanding, as assessed by comprehension scores, was unrelated to OC tendencies. LimitationsAmong other limitations discussed in the article, our results were obtained on a non-selected sample of students varying on OC tendencies. It would be important to replicate these findings with diagnosed OCD participants as compared to both non-clinical and anxiety disorders control participants. ConclusionsThese findings extend the SPIS model to the domain of understanding and may have important clinical implications.

Electrotactile Feedback in a Virtual Hand Rehabilitation Platform: Evaluation and Implementation

Tactile feedback plays an important role in hand manipulation, especially in the grasping process which is one of the major functions of the hand. However, few commercially available prosthetic hands or hand motor function rehabilitation systems are equipped with tactile feedback. The absence of suitable tactile feedback modules leads to an inferior rehabilitation performance with a large burden on user training and compromised usability. Thus, it is challenging and essential to integrate a proper tactile feedback module with the existing hand rehabilitation systems to achieve a better control performance and accelerate the rehabilitation process. This paper focuses on the implementation and evaluation of the electrotactile feedback (EF) enhanced rehabilitation system. A virtual hand rehabilitation platform is proposed comprising an surface electromyography (sEMG) acquisition module, an electrotactile stimulation module, a virtual environment with sEMG-driven humanlike hand and numerical feedbacks of grasping force and fingertip deformation, where a closed-loop control is formed. Three different feedback conditions including visual feedback (VF), EF, and no feedback (NF) are compared based on the proposed platform. Experiments were conducted on 10 able-bodied subjects, and multiple quantitative metrics for the rehabilitation performance evaluation including training burden estimation and success rate (SR) of tasks were adopted. Results indicate that the integration of EF is helpful to both reduce the rehabilitation duration and improve the virtual grasping SR in comparison with the NF condition while possessing a better practicality over VF. Note to Practitioners —This paper is motivated by the problem of hand grasp control for rehabilitation purposes, but it also applies to other hand motor function rehabilitation process. Existing hand motor function rehabilitation approaches generally lack a proper feedback and rely on the heavy burden during user training and the users experience. This paper suggests incorporating EF to improve the efficiency and efficacy of the rehabilitation process. The electrical stimulation is driven by the myoelectric-sensing-based force estimation and encoded in a manual scheme to fit each individual involved. In our work, a virtual hand rehabilitation platform is implemented to verify the feasibility of EF in reducing the burden of user training and improving the rehabilitation performance, which allows the expanding of the current system into a broader spectrum of motor function rehabilitation applications. Experiments on able-bodied subjects suggest that the EF in the proposed virtual hand rehabilitation platform is feasible, but it has not been tested on the limb-impaired subjects and confined to a manual encoding of electrical stimulation. In the future research, the design of a general EF enhanced hand rehabilitation platform with a standardized stimulation parameter optimization will be addressed and further validated on the subjects with limb impairments and amputation.

Percolation of interdependent networks with conditional dependency clusters

Modern systems are always coupled. Previous studies indicate that coupled systems are more fragile than single systems. In a single system, when a fraction of 1-<i>p</i> nodes are removed, the percolation process is often of the second order. In a coupled system, due to the lack of resilience, the phase transition is always of the first order when removing a fraction of nodes. Most of previous studies on coupled systems focus on one-to-one dependency relation. This kind of relationship is called a no-feedback condition. Existing studies suppose that coupled systems are much more fragile without a no-feedback condition. That is to say, if a node depends on more than one node, the coupled system will breakdown even when a small fraction of nodes are removed from the coupled system. By observing the real world system, real nodes are often dependent on a dependency cluster, which consists of more than one other node. For example, in an industry chain, an electronic equipment factory may need several raw material factories to supply production components. Despite part of the raw material factories being bankrupt, the electronic equipment factory can carry out productionnormally because the remaining raw material factories still supply the necessary production components. But theoretical analysis shows that the robustness of such a coupled system is worse than that of one-to-one dependency system. Actually, the coupled system in real world does not usually disintegrate into pieces after some nodes have become invalid. To explain this phenomenon, we model a coupled system as interdependent networks and study, both analytically and numerically, the percolation in interdependent networks with conditional dependency clusters. A node in our model survives until the number of failed nodes in its dependency cluster is greater than a threshold. Our exact solutions of giant component size are in good agreement with the simulation results. Though our model does not have second order phase transition, we still find ways to improve the robustness of interdependent networks. One way is to increase the dependency cluster failure threshold. A higher threshold means that more nodes in the dependency cluster can be removed without breaking down the node depending on the cluster. Other way is to increase the size of dependency clusters, the more the nodes in the dependency cluster, the more the failure combinations are, which increases the survival probability of the node depending on cluster. Our model offers a useful strategy to enhance the robustness of coupled system and makes a good contribution to the study of interdependent networks with dependency clusters.

Respiratory Biofeedback Does Not Facilitate Lowering Arousal in Meditation Through Virtual Reality

The current study examined the effectiveness of respiratory biofeedback in lowering subjective and objective arousal after stress. Participants were presented with a meditation session in virtual reality while subjective and objective arousal were measured, the latter measured through ECG and EEG. Three conditions were used: (a) a respiratory biofeedback condition, in which visual feedback was paired to breathing; (b) a control feedback placebo condition, in which visual feedback was not paired to breathing; and (c) a control no-feedback condition, in which no visual feedback was used. Subjective and objective arousal decreased during meditation after stress in all conditions, demonstrating recovery after stress during meditation in virtual reality. However, the reduction in arousal (on all outcome measures combined and heart rate specifically) was largest in the control feedback placebo condition, in which no biofeedback was used, indicating that respiratory biofeedback had no additional value in reducing arousal. The findings of the current study highlight the importance of including a control feedback placebo condition in order to establish the exact additional value of biofeedback and offer insights in applying cost-effective virtual reality meditation training.

Factors underlying risk taking in heroin-dependent individuals: Feedback processing and environmental contingencies

Evidence suggests that factors influencing risk-taking include whether decisions are made based on emotions (affective systems) or cognitions (deliberative systems), the processing of feedback (e.g., deciding to attend a rehabilitation facility for opioid addiction treatment after an intervention held by a family member), and attention to environmental contingencies (e.g., considering the probability of an outcome such as the likelihood of contracting tetanus from a shared needle; or the gains and losses associated with a decision, such as the benefits and costs of taking drugs). Although drug-dependent individuals tend to take more risks than non-drug users, the factors underlying risk-taking are unknown. The current study tested, for the first time, the influences of performance feedback (i.e., whether feedback about performance is integrated into decision-making in heroin-dependent individuals) and attention to environmental contingencies (i.e., the influence of the probability of a loss, the gain amount, and the loss amount associated with a scenario) on risk-taking in heroin-dependent individuals. Heroin-dependent patients undergoing maintenance therapy for opioid addiction (n = 25) and healthy controls (n = 27) completed the feedback and no-feedback conditions of the Columbia Card Task (CCT). Analyses of covariance, controlling for education and task design (the order in which the CCT conditions were completed) as covariates revealed a significant interaction between (a) probability, gain and loss amount, and group, and (b) group and probability. Our findings suggest that heroin-dependent patients pay less attention to environmental contingencies during risk-taking than controls. Addressing these factors may facilitate greater adherence to treatment programs and lower rates of relapse.

Secondary Confessions as Post-identification Feedback: How Jailhouse Informant Testimony Can Alter Eyewitnesses’ Identification Decisions

Prior research has shown that primary confession evidence can alter eyewitnesses’ identifications and self-reported confidence. The present study investigated whether secondary confession evidence from a jailhouse informant could have the same effect. Participants (N = 368) watched a video of an armed robbery and made an identification decision from a photo lineup. Except for those in the no-feedback conditions, all participants then read that certain lineup members either confessed to the crime, denied involvement, or were implicated by a jailhouse informant. Jailhouse informant testimony implicating the identified lineup member led participants to have significantly higher confidence in their identification. In contrast, jailhouse informant testimony that implicated a lineup member other than the identified led participants to have significantly lower confidence in their initial identification, and 80% of these witnesses changed their identification. These results indicate that jailhouse informant testimony can influence eyewitnesses’ confidence and their identification decisions.

Monetary Reward and Punishment to Response Inhibition Modulate Activation and Synchronization Within the Inhibitory Brain Network.

A reward or punishment can modulate motivation and emotions, which in turn affect cognitive processing. The present simultaneous functional magnetic resonance imaging-electroencephalography study examines neural mechanisms of response inhibition under the influence of a monetary reward or punishment by implementing a modified stop-signal task in a virtual battlefield scenario. The participants were instructed to play as snipers who open fire at a terrorist target but withhold shooting in the presence of a hostage. The participants performed the task under three different feedback conditions in counterbalanced order: a reward condition where each successfully withheld response added a bonus (i.e., positive feedback) to the startup credit, a punishment condition where each failure in stopping deduced a penalty (i.e., negative feedback), and a no-feedback condition where response outcome had no consequences and served as a control setting. Behaviorally both reward and punishment conditions led to significantly down-regulated inhibitory function in terms of the critical stop-signal delay. As for the neuroimaging results, increased activities were found for the no-feedback condition in regions previously reported to be associated with response inhibition, including the right inferior frontal gyrus and the pre-supplementary motor area. Moreover, higher activation of the lingual gyrus, posterior cingulate gyrus (PCG) and inferior parietal lobule were found in the reward condition, while stronger activation of the precuneus gyrus was found in the punishment condition. The positive feedback was also associated with stronger changes of delta, theta, and alpha synchronization in the PCG than were the negative or no-feedback conditions. These findings depicted the intertwining relationship between response inhibition and motivation networks.

Does the accuracy matter? Accurate concept map feedback helps students improve the cohesion of their explanations

Students are often challenged by the demand of writing cohesive explanatory texts. Prior research has shown that providing students with concept map feedback that visualizes explanatory cohesion deficits helped students generate more cohesive explanations. We conducted an experiment to investigate whether the accuracy of the provided information within the concept map feedback affected students’ improvements of cohesion. Accordingly, we varied the represented accuracy of information within such concept maps: Students either received accurate concept map feedback that depicted the real relations between concepts, as well as the authentic cohesion gaps in their explanations, or students received inaccurate concept map feedback, which depicted randomly drawn relations and random cohesion gaps. Additionally, in a baseline condition, students did not receive any feedback. We found that the students in the accurate feedback condition generated more cohesive explanations than the students in the no-feedback condition, whereas the students in the inaccurate feedback condition lay in-between. Evidently, providing feedback in general can be regarded as beneficial to enhance students’ writing. However, the accuracy of the provided feedback further impacts the effectiveness of computer-generated concept maps.

Learning effects of dynamic postural control by auditory biofeedback versus visual biofeedback training

Augmented sensory biofeedback (BF) for postural control is widely used to improve postural stability. However, the effective sensory information in BF systems of motor learning for postural control is still unknown. The purpose of this study was to investigate the learning effects of visual versus auditory BF training in dynamic postural control. Eighteen healthy young adults were randomly divided into two groups (visual BF and auditory BF). In test sessions, participants were asked to bring the real-time center of pressure (COP) in line with a hidden target by body sway in the sagittal plane. The target moved in seven cycles of sine curves at 0.23Hz in the vertical direction on a monitor. In training sessions, the visual and auditory BF groups were required to change the magnitude of a visual circle and a sound, respectively, according to the distance between the COP and target in order to reach the target. The perceptual magnitudes of visual and auditory BF were equalized according to Stevens’ power law. At the retention test, the auditory but not visual BF group demonstrated decreased postural performance errors in both the spatial and temporal parameters under the no-feedback condition. These findings suggest that visual BF increases the dependence on visual information to control postural performance, while auditory BF may enhance the integration of the proprioceptive sensory system, which contributes to motor learning without BF. These results suggest that auditory BF training improves motor learning of dynamic postural control.

Informationally administered reward enhances intrinsic motivation in schizophrenia

Even when individuals with schizophrenia have an intact ability to enjoy rewarding moments, the means to assist them to translate rewarding experiences into goal-directed behaviors is unclear. The present study sought to determine whether informationally administered rewards enhance intrinsic motivation to foster goal-directed behaviors in individuals with schizophrenia (SZ) and healthy controls (HCs). Eighty-four participants (SZ=43, HCs=41) were randomly assigned to conditions involving either a performance-contingent reward with an informationally administered reward or a task-contingent reward with no feedback. Participants were asked to play two cognitive games of equalized difficulty. Accuracy, self-reported intrinsic motivation, free-choice intrinsic motivation (i.e., game play during a free-choice observation period), and perceived competency were measured. Intrinsic motivation and perceived competency in the cognitive games were similar between the two participant groups. The informationally administered reward significantly enhanced self-reported intrinsic motivation and perceived competency in both the groups. The likelihood that individuals with schizophrenia would play the game during the free-choice observation period was four times greater in the informationally administered reward condition than that in the no-feedback condition. Our findings suggest that, in the context of cognitive remediation, individuals with schizophrenia would benefit from informationally administered rewards.

Learner affect in computerised L2 oral grammar practice with corrective feedback

ABSTRACTAlthough corrective feedback (CF) has received much interest in the second language acquisition literature, relatively little research has investigated the relationship between CF and learner affect in concrete practice situations. The present study investigates learners’ affective states and practice behaviour in a novel context: oral grammar practice with a computer-assisted language learning (CALL) system employing automatic speech recognition (ASR) technology to analyse learners’ speech and provide feedback. Thirty-one adult learners of Dutch practiced with this system in one of two conditions: the no-feedback condition (NOCF) and the feedback condition (CF) which provided immediate CF through ASR. Despite concerns that CF can elicit negative affective reactions and although practice with feedback forced learners to reformulate more often, CF did not appear to have a negative impact. Our analysis finds no significant differences between the NOCF and CF groups. A significant correlation between practice performance and self-efficacy was found in the CF only. These findings suggest that ASR-enabled CALL systems may be suitable environments for oral grammar practice where CF on oral productions can be provided without negative affective responses, and that without feedback, learners may develop self-efficacy beliefs which do not necessarily reflect their actual performance.

The Effects of Positive and Negative Feedback on Maximal Voluntary Contraction Level of the Biceps Brachii Muscle: Moderating Roles of Gender and Conscientiousness.

We investigated the effect of positive and negative feedback on maximal voluntary contraction (MVC) of the biceps brachii muscle and explored the mediating effects of gender and conscientiousness. During elbow flexion, MVCs were measured in positive, negative, and no-feedback conditions. Participants were divided into high- and low-conscientiousness groups based on the median split of their scores on Tatar's five-factor personality inventory. Considering all participants 46 college student athletes (21 female, 28 male), positive feedback led to a greater MVC percentage change (-5.76%) than did negative feedback (2.2%). MVC percentage change in the positive feedback condition differed significantly by gender, but the negative feedback condition did not. Thus, positive feedback increased female athletes' MVC level by 3.49%, but decreased male athletes' MVC level by 15.6%. For conscientiousness, MVC percentage change in the positive feedback condition did not differ according to high and low conscientiousness. However, conscientiousness interacted with gender in the positive feedback condition, increasing MVC in high-conscientiousness female athletes and decreasing MVC in low-conscientiousness female athletes. Positive feedback decreased MVC in both high- and low-conscientiousness male athletes.

Similarities between adult female crack cocaine users and adolescents in risky decision-making scenarios

ABSTRACTBackground: Although previous studies have shown that both adolescence and drug addiction can influence risk-taking and decision-making processes, the underlying mechanisms remain unclear. Specifically, there is a lack of evidence as to whether these conditions could affect deliberative and affective processes involved in risk taking, such as feedback learning and valuation of profits and risk. Objectives: The objectives were to compare the role of feedback and the use of information in risk-taking behavior between female crack cocaine users and adolescents. Additionally, we aimed to investigate whether sensation seeking, impulsivity, depressive and anxiety symptoms, executive functioning, and working memory performance could explain differences in risk-taking behavior. Method: This is a quasi-experimental study comparing 27 low-income adult female crack cocaine users (CU) to 18 female adolescents (AD) within two conditions (no-feedback or delayed-feedback) of the Columbia Card Task (CCT). In order to investigate CCT reference values for adult females, we also included 20 female non-drug-users with regular education and income as a reference group (RG). Results: A similar pattern of risk-taking behavior was found between CU and AD within the CCT no-feedback condition. When delayed feedback was provided, AD exhibited a similar pattern of risk-taking behavior in the no-feedback condition, while CU showed a reduction of risk-taking behavior. Both groups exhibited higher risk taking than the RG within the CCT no-feedback condition, but only the AD group showed higher risk-taking behavior within the CCT feedback condition. Depressive symptom severity and working memory deficits were associated with higher risk-taking behaviors in CU. Executive functioning deficits were associated with higher risk-taking behavior in AD. Conclusions: Adult female crack cocaine users and female adolescents took similar risks during risky decision-making scenarios where feedback about their own performance was absent. However, when participants were provided with such feedback, it modulated risk-taking behaviors in crack cocaine users but not in adolescents.

The Role of Informative and Ambiguous Feedback in Avoidance Behavior: Empirical and Computational Findings

Avoidance behavior is a critical component of many psychiatric disorders, and as such, it is important to understand how avoidance behavior arises, and whether it can be modified. In this study, we used empirical and computational methods to assess the role of informational feedback and ambiguous outcome in avoidance behavior. We adapted a computer-based probabilistic classification learning task, which includes positive, negative and no-feedback outcomes; the latter outcome is ambiguous as it might signal either a successful outcome (missed punishment) or a failure (missed reward). Prior work with this task suggested that most healthy subjects viewed the no-feedback outcome as strongly positive. Interestingly, in a later version of the classification task, when healthy subjects were allowed to opt out of (i.e. avoid) responding, some subjects (“avoiders”) reliably avoided trials where there was a risk of punishment, but other subjects (“non-avoiders”) never made any avoidance responses at all. One possible interpretation is that the “non-avoiders” valued the no-feedback outcome so positively on punishment-based trials that they had little incentive to avoid. Another possible interpretation is that the outcome of an avoided trial is unspecified and that lack of information is aversive, decreasing subjects’ tendency to avoid. To examine these ideas, we here tested healthy young adults on versions of the task where avoidance responses either did or did not generate informational feedback about the optimal response. Results showed that provision of informational feedback decreased avoidance responses and also decreased categorization performance, without significantly affecting the percentage of subjects classified as “avoiders.” To better understand these results, we used a modified Q-learning model to fit individual subject data. Simulation results suggest that subjects in the feedback condition adjusted their behavior faster following better-than-expected outcomes, compared to subjects in the no-feedback condition. Additionally, in both task conditions, “avoiders” adjusted their behavior faster following worse-than-expected outcomes, and treated the ambiguous no-feedback outcome as less rewarding, compared to non-avoiders. Together, results shed light on the important role of ambiguous and informative feedback in avoidance behavior.

Do I really feel it? The contributions of subjective fluency and compatibility in low-level effects on aesthetic appreciation.

The causes for the liking of objects are multifaceted. According to the processing fluency account, the ease with which an object is processed leads to a subjective feeling of fluency. This subjective feeling is then interpreted as a positive reaction toward the object resulting higher liking. However, evidence regarding the processes underlying this relation is scarce. To show that the subjective feeling can indeed be responsible for liking, we experimentally manipulated processing ease by providing false physiological feedback (varying skin conductance indicated varying feelings of fluency) and by varying presentation times between 100 and 400 ms while participants viewed line drawings of objects and rated them for liking. A first experiment showed that both false physiological feedback and presentation duration influenced liking. Stimuli primed with a (fake) visualization of a physiological correlate of high ease of processing were liked more than stimuli primed with a low ease of processing. Liking ratings in a no-feedback condition fell between the high and low feedback conditions. To explore possible compatibility effects of coupling visual feedback to the fluency interpretation, in a second experiment we reversed the feedback interpretation—visualization of high skin conductance now indicated low ease of processing. The results show a similar pattern, though the effect was subtler. This indicates that when the coupling of feedback to fluency is less apparent or less compatible, the feeling is less strongly linked to liking. Our results support the claim that variations in the feeling of fluency affect the appreciation of objects in terms of liking. Together, the experiments suggest the contributions of processing ease as well as compatibility to the experience of liking.

The Impact of Different Degrees of Feedback on Physical Activity Levels: A 4-Week Intervention Study.

Assessing levels of physical activity (PA) and providing feedback about these levels might have an effect on participant’s PA behavior. This study discusses the effect of different levels of feedback—from minimal to use of a feedback display and coach—on PA over a 4-week intervention period. PA was measured at baseline, during and immediately after the intervention. Participants (n = 227) were randomly assigned to a Minimal Intervention Group (MIG-no feedback), Pedometer Group (PG-feedback on steps taken), Display Group (DG-feedback on steps, minutes of moderate to vigorous physical activity and energy expenditure) or Coaching Group (CoachG-same as DG with need-supportive coaching). Two-way ANCOVA showed no significant Group × Time interaction effect for the different PA variables between the MIG and PG. Also no differences emerged between PG and DG. As hypothesized, CoachG had higher PA values throughout the intervention compared with DG. Self-monitoring using a pedometer resulted in more steps compared with a no-feedback condition at the start of the intervention. However, adding individualized coaching seems necessary to increase the PA level until the end of the intervention.

Stimulus-dependent modulation of visual neglect in a touch-screen cancellation task.

Patients with left-sided neglect frequently show omissions and repetitive behavior on cancellation tests. Using a touch-screen-based cancellation task, we tested how visual feedback and distracters influence the number of omissions and perseverations. Eighteen patients with left-sided visual neglect and 18 healthy controls performed four different cancellation tasks on an iPad touch screen: no feedback (the display did not change during the task), visual feedback (touched targets changed their color from black to green), visual feedback with distracters (20 distracters were evenly embedded in the display; detected targets changed their color from black to green), vanishing targets (touched targets disappeared from the screen). Except for the condition with vanishing targets, neglect patients had significantly more omissions and perseverations than healthy controls in the remaining three subtests. Both conditions providing feedback by changing the target color showed the highest number of omissions. Erasure of targets nearly diminished omissions completely. The highest rate of perseverations was observed in the no-feedback condition. The implementation of distracters led to a moderate number of perseverations. Visual feedback without distracters and vanishing targets abolished perseverations nearly completely. Visual feedback and the presence of distracters aggravated hemispatial neglect. This finding is compatible with impaired disengagement from the ipsilesional side as an important factor of visual neglect. Improvement of cancellation behavior with vanishing targets could have therapeutic implications.

Feedback May Harm: Role of Feedback in Probabilistic Decision Making of Adolescents with ADHD.

Inept probabilistic decision making is commonly associated with ADHD. In experimental designs aimed to model probabilistic decision making in ADHD, feedback following each choice was, in the majority of studies, part of the paradigm. This study examined whether feedback processing plays a role in the maladaptive choice behavior of subjects with ADHD by comparing feedback and no-feedback conditions. Sixty adolescents (49 males), ages 13-18, with and without ADHD, performed a descriptive probabilistic choice task in which outcomes and probabilities were explicitly provided. Subjects performed the task either with or without feedback. Under the no-feedback condition, adolescents with ADHD and controls performed similarly, whereas under the feedback condition, subjects with ADHD chose the unfavorable outcomes more frequently and risked smaller sums than controls. These finding demonstrate the crucial role of feedback in the decision making of adolescents with ADHD.

Percolation on interacting networks with feedback-dependency links.

When real networks are considered, coupled networks with connectivity and feedback-dependency links are not rare but more general. Here, we develop a mathematical framework and study numerically and analytically the percolation of interacting networks with feedback-dependency links. For the case that all degree distributions of intra- and inter- connectivity links are Poissonian, we find that for a low density of inter-connectivity links, the system undergoes from second order to first order through hybrid phase transition as coupling strength increases. It implies that the average degree k of inter-connectivity links has a little influence on robustness of the system with a weak coupling strength, which corresponds to the second order transition, but for a strong coupling strength corresponds to the first order transition. That is to say, the system becomes robust as k increases. However, as the average degree k of each network increases, the system becomes robust for any coupling strength. In addition, we find that one can take less cost to design robust system as coupling strength decreases by analyzing minimum average degree kmin of maintaining system stability. Moreover, for high density of inter-connectivity links, we find that the hybrid phase transition region disappears, the first order region becomes larger and second order region becomes smaller. For the case of two coupled scale-free networks, the system also undergoes from second order to first order through hybrid transition as the coupling strength increases. We find that for a weak coupling strength, which corresponds to the second order transitions, feedback dependency links have no effect on robustness of system relative to no-feedback condition, but for strong coupling strength which corresponds to first order or hybrid phase transition, the system is more vulnerable under feedback condition comparing with no-feedback condition. Thus, for designing resilient system, designers should try to avoid the feedback dependency links, because the existence of feedback-dependency links makes the system extremely vulnerable and difficult to defend.

Robustness of network of networks with interdependent and interconnected links

Robustness of network of networks (NON) has been studied only for dependency coupling (Gao et al., 2012) and only for connectivity coupling (Leicht and Souza, arxiv:0907.0894). The case of network of networks with both interdependent and interconnected links is more complicated, and also more close to real-life coupled network systems. Understanding the robustness of NON with interdependent and interconnected links is helpful to design resilient infrastructures. Here we develop a framework to study analytically and numerically the robustness of this system with no-feedback and feedback conditions for the case of starlike NON. When assumed that all degree distributions of the connectivity intra- and inter-links are Poissonian, we find that the system undergoes from second order through hybrid order to first order phase transition as coupling strength q increases. Additionally, for both conditions, the results suggest that increasing density of connectivity links (intra-connectivity links or inter-connectivity links) can increase the robustness of the system, while the interdependency links decrease its robustness. Furthermore, by comparing critical attacking strength under same parameters for both conditions, we also find that feedback condition of dependency links, in contrast to no-feedback condition, makes the system extremely vulnerable. Although our detailed analysis is for Poisson degree distribution, the theory can be applied to any degree distribution and other basic topological structure of network such like tree-like and loop-like NON.