"A mathematical fractal is generated by an infinitely recursive process, in which the final level of detail is never reached, and never can be reached by increasing the scale at which observations are made. In reality, fractals are generated by finite processes, and exhibit no visible change in detail after a certain resolution limit. This behavior of natural fractal objects is similar to the exponential cutoff, which can be observed in many degree distributions of real networks." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"As opposed to engineered systems, evolutionary networks are integrated and their parts cannot be optimized separately." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Creative elements add random elements to network behavior, inducing an increase in noise. This is highly beneficial to a certain extent as we saw in the previous box, but becomes intolerable if it exceeds a certain threshold. This threshold is high if the hosting network lives an individual life and often meets unexpected situations. However, the same threshold becomes low if the hosting network is part of a higher level organization which provides a stable environment." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Democracy is a highly robust, but at the same time very fragile, complex system, which requires constant change to maintain its self-organized state. Democracy is by definition not in equilibrium. The random graph pattern may pose a system of boring and very low-complexity equilibrium. If all of us ever find ourselves with plenty of resources, that will be THE END of our history." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Fractals are self-similar objects. However, not every self-similar object is a fractal, with a scale-free form distribution. If we put identical cubes on top of each other, we get a self-similar object. However, this object will not have scale-free statistics: since it has only one measure of rectangular forms, it is single-scaled. We need a growing number of smaller and smaller self-similar objects to satisfy the scale-free distribution." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"If a network has violently changing properties, it is most probably not very stable. How can we measure stability, if a network remains unchanged? The assessment of stability often requires a test, and this test comes in the form of a perturbation to the network. A stable network should try to restore its original status after a perturbation. However, this is not easy. Most networks are open systems and therefore undergo a continuous series of perturbations." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Network stability may be a key element in the development of multilevel, nested networks. The formation of nested networks obviously requires at least a few contacts between the bottom networks. However, evolutionary selection requires the independence and at least temporary isolation of the bottom networks themselves. Weak links are probably the only tools for solving this apparent paradox." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Noise is bad for the network, if high and continuous noise levels disturb all network functions. So far, the take-home message is that we have to stop noise in order to survive. This assumption is wrong. Reducing the noise to zero would mean no interaction of the network with the environment. Isolation is clearly a bad strategy, since such an isolated network will die. However, zero noise is bad for another reason too. Noise can be helpful in many ways. The first documented observations of good noise were sailors’ reports on the peculiar phenomenon that disordered raindrops falling on the ocean can calm roughseas. Another example of the optimal level of noise is opinion formation. A low noise is not enough for modulation of opinion formation, while strong fluctuations prevent the formation of a definitive collective opinion." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Perturbations are often regarded as noise. What is the difference? Noise is usually understood from the point of the experimenter. If we measure it from the outside, noise is the fluctuation of the value we measure. However, from the point of view of the network, noise is a series ofperturbations changing its original status. Network perturbations can be called either signals or noise." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Practically every complex system can be imagined as a network. Atoms form a network making macromolecules. Proteins form a network making cells. Cells form a network making organs and bodies. We form a network making our societies, and so on. Most of these networks are a result of self-organization. In fact, self-organization seems to be an inherent property of matter in our Universe. The resulting networks have a lot of common features, from their topology to their dynamism." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Scale-free degree distribution is not always the optimal solution to the requirement of cost efficiency. As mentioned before, in smallworld networks, building and maintaining links between network elements requires energy. Therefore, network topology is a result of an optimization process. It is optimized with respect to the available resources to ensure optimal communication between different network parts. If the network enjoys unlimited resources, it will have a random distribution with plenty of links. Scale-free degree distribution occurs as a result of optimization in systems with finite resources. If the network experiences even more limited resources, the degree distribution will be steeper than in the case of a scale-free network, and a transition will therefore occur towards a star network. These phenomena are called topological phase transitions [...]" (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Scale-free topologies enable more sensitive responses to various changes than those allowed by random networks (Bar-Yam and Epstein, 2004). This can be a very important property for explaining why scale-free networks have been selected and maintained in many systems." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Self-organizing networks suffer various types of random damage. Therefore, if the network remained static, it would soon become dysfunctional. Some networks have developed highly specific screening systems which recognize and repair random damage. On the one hand, this process requires energy, which arrives in the form of perturbations or noise. On the other hand, noise-triggered network restructuring will repeat a few steps of the original self-organization and therefore constitutes a much cheaper way of providing a continuous repair function, with the additional advantage that it is always adaptive with respect to the actual environment of the network." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"The scale-free distribution pattern has been most studied on the degree distribution of networks. What is a degree? The degree of a network. element is the number of connections it has. A scale-free degree distribution means that the network has a large number of elements with very few neighbors, but it has a non-zero number of elements with an extraordinarily large number of neighbors. These connection-rich elements are called hubs. If an element has just a few connections, it is often called a node." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"The scale-free system is a ‘borderline’ case between the random graph and the star phase. It is very fragile and transient, but in spite of this, it is very robust. We call it democracy. This democracy net always keeps a delicate balance betweenanarchy (random net) and dictatorship (star net). Fortunately, in democratic systems, the society is not segmented and weak links flourish. Consequently, the fragile system becomes robust. Democratic systems show the greatest complexity of all. However, weak links and their buffering may grow too great. A democratic society remains flexible for smaller challenges, but occasionally may become overcomplicated and unable to make a fast response to a life-threatening danger." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"To understand, how noise is related to scale-freeness, we have to do some mathematics again. Noise is usually characterized by a mathematical trick. The seemingly random fluctuation of the signal is regarded as a sum of sinusoidal waves. The components of the million waves giving the final noise structure are characterized by their frequency. To describe noise, we plot the contribution (called spectral density) of the various waves we use to model the noise as a function of their frequency. This transformation is called a Fourier transformation [...]" (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Weak links are both elusive and overwhelming. Science has grown used, to examining strong links. Strong links are always there. Strong links are reproducible. Strong links are few in number and hence comprehensible. Strong links are already known. Strong links are scientific. Strong links are exciting. In short, strong links are like friends to us. In contrast, weak links are transient. Weak links are undetectable. Weak links are overwhelmingly numerous. Weak links are unknown. Weak links are unscientific. Weak links are hopeless. In short, weak links are like foes to us."(Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"Weak links are links between network elements, which connect them with a low intensity. Weak links may also connect network elements with a higher intensity, but in this case they are only transient. [...] A link is defined as weak when its addition or removal does not change the mean value of a target measure in a statistically discernible way." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
"[...] weak links seem to be necessary for the development of small-worldness, are a consequence of scale-freeness, and make a key contribution to the formation of nestedness. Weak links are also general and important elements of networks, forming most of their contacts. When we talk about the reason why we like networks, we have to talk about weak-linkedness." (Péter Csermely, "Weak Links: The Universal Key to the Stabilityof Networks and Complex Systems", 2009)
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