Albert-László Barabási

Albert-László Barabási, Linked: The New Science of Networks, Cambridge, Mass., Perseus Pub., 2002.

The construction and structure of graphs or networks is the key to understanding the complex world around us. Small changes in the topology, affecting only a few of the nodes or links, can open up hidden doors, allowing new possibilities to emerge. [12]

Random network theory tells us that as the average number of links per node increases beyond the critical one, the number of nodes left out of the giant cluster decreases exponentially. That is, the more links we add, the harder it is to find a node that remains isolated. Nature does not take risks by staying close to the threshold. It well surpasses it. Consequently, the networks around us are not just webs. They are very dense networks from which nothing can escape and within which every node is navigable. [19]

Stanley Milgram awakened us to the fact that not only are we connected, but we live in a world in which no one is more than a few handshakes from anyone else. That is, we live in a small world. Our world is small because society is a very dense web. We have far more friends than the critical one needed to keep us connected. [30]

The number of social links an individual can actively maintain has increased dramatically, bringing down the degrees of separation. Stanley Milgram estimated six. Frigyes Karinthy five. We could be much lcoser these days to three. [39]

Our ability to reach people has less and less to do with the physical distance between us. Discovering common acquaintances with perfect strangers on worldwide trips repeatedly reminds us that some people on the other side of the planet are often closer along the social network than people living next door. Navigating this non-Euclidean world repeatedly tricks our intuition and reminds us that there is a new geometry out there that we need to master in order to make sense of the complex world around us. [40]

Hubs appear in most large complex networks that scientists have been able to study so far. They are ubiquitous, a generic building block of our complex, interconnected world. [63]

The power law distribution thus forces us to abandon the idea of a scale, or a characteristic node. In a continuous hierarchy there is no single node which we could pick out and claim to be characteristic of all the nodes. There is no intrinsic scale in these networks. ... With the realization that most complex networks in nature have a power-law degree distribution, the term scale-free networks rapidly infiltrated most disciplines faced with complex webs. [70]

Nature normally hates power laws. In ordinary systems all quantities follow bell curves, and correlations decay rapidly, obeying exponential laws. But all that changes if the system is forced to undergo a phase transition. Then power laws emerge - nature's unmistakable sign that chaos is departing in favor of order. ... They are the patent signatures of self-organization in complex systems. [77]

The scale-free topology is a natural consequence of the ever-expanding nature of real networks. ... Thanks to growth and preferential attachment, a few highly connected hubs emerge. [87]

Growth and preferential attachment can explain the basic features of the networks seen in nature. [91]

In networks that display fit-get-rich behavior, competition leads to a scale-free topology. Most networks we have studied so far - the Web, the Internet, the cell, Hollywood, and many other real networks - belong to this category. The winner shares the spotlight with a continuous hierarchy of hubs.
Yet Bose-Einstein condensation offers the theoretical possibility that in some systems the winner can grab all the links. When that happens, the scale-free topology vanishes. [107]

Most systems displaying a high degree of tolerance against failures share a common feature: Their functionality is guaranteed by a highly interconnected complex network. ... It seems that nature strives to achieve robustness through interconnectivity. Such universal choice of a network architecture is perhaps more than mere coincidence. [111]

Similarly, in scale-free networks, failures predominantly affect the numerous small nodes. Thus, these networks do not break apart under failures. The accidental removal of a single hub will not be fatal either, since the continuous hierarchy of several large hubs will maintain the network's integrity. Topological robustness is thus rooted in the structural unevenness of scale-free networks: Failures disproportionately affect small nodes. [114]

For scale-free networks the critical threshold to break break apart disappears in cases where the degree exponent is smaller or equal to three. Amazingly, most networks of interest, ranging from the Internet to the cell, are scale-free and have a degree exponent smaller than three. Therefore, these networks break apart only after all nodes have been removed - or, for all practical purposes, never. [115]

The response of scale-free networks to attacks is similar to the behavior of random networks under failures. There is a crucial difference, however. We do not need to remove a large number of nodes to reach the critical point. Disable a few of the hubs and a scale-free network will fall to pieces in no time. [117]

The removal of the most connected nodes rapidly disintegrates these networks, breaking them into tiny noncommunicating islands. Therefore, hidden within their structure, scale-free networks harbor an unsuspected Achilles' heel, coupling a robustness against failures with vulnerability to attack. [118]

When a network acts as a transportation system, a local failure shifts loads or responsibilities to other nodes. If the extra load is negligible, it can be seamlessly absorbed by the rest of the system, and the failure remains effectively unnoticed. If the extra load is too much for the neighboring nodes to carry, they will either tip or again redistribute the load to their neighbors. Either way, we are faced with a cascading failure. [120]

Most cascades are not instantaneous: Failures can go unnoticed for a long time before starting a landslide. Attempting to decrease the frequency of such cascades has inevitable consequences, however, as those cascades that do succeed are then more disruptive. [121]

The Pfizer study of how physicians adopt a new drug demonstrated that innovations spread from innovators to hubs. The hubs in turn send the information out along their numerous links, reaching most people within a given social or professional network. [129]

In scale-free networks the epidemic threshold miraculously vanished! That is, even if a virus is not very contagious, it spreads and persists. Defying all wisdom accumulated during five decades of diffusion studies, viruses traveling in scale-free networks do not appear to notice any threshold. They are practically unstoppable. [135]

While you could persuade an institution to close down the portion of the network under its authority, no single company or person controls more than a negligible fraction of the whole Internet. The underlying network has become so distributed, decentralized, and locally guarded that even such an ordinary task as getting a central map of it has become virtually impossible. [148]

While entirely of human design, the Internet now lives a life of its own. It has all the characteristics of a complex evolving system, making it more similar to a cell than a computer chip. Many diverse components, developed separately, contribute to the functioning of a system that is far more than the sum of its parts. [150]

Like architects' buildings, the Web's architecture is the product of two equally important layers: code and collective human actions taking advantage of the code. The first can be regulated by courts, government, and companies alike. The second, however, cannot be shaped by any single user or institution, because the Web has no central design - it is self-organized. It evolves from the individual actions of millions of users. As a result, its architecture is much richer than the sum of its parts. Most of the Web's truly important features and emerging properties derive from its large-scale self-organized topology. [174]

As research, innovation, product development, and marketing become more and more specialized and divorced from each other, we are converging to a network economy in which strategic alliances and partnerships are the means for survival in all industries. [208]

In a network economy, buyers and suppliers are not competitors but partners. The relationship between them is often very long lasting and stable.
The stability of these links allows companies to concentrate on their core business. If these partnerships break down, the effects can be severe. Most of the time failures handicap only the partners of the broken link. Occasionally, however, they send ripples through the whole economy. [209]

In reality, the market is nothing but a directed network. Companies, firms, corporations, financial institutions, governments, and all potential economic players are the nodes. Links quantify various interactions between these institutions, involving purchases and sales, joint research and marketing projects, and so forth. The weight of the links captures the value of the transaction, and the direction points from the provider to the receiver. The structure and evolution of this weighted and directed network determine the outcome of all macroeconomic processes. [209]

If we view the economy as a highly interconnected network of companies and financial institutions, we can begin to make sense of these events. In such networks the failure of a node has little effect on the system's integrity. Occasionally, however, the breakdown of some well-selected nodes sets off a cascade of failures that can shake the whole system. [211]

Cascading failures are a direct consequence of a network economy, of interdependencies induced by the fact that in a global economy no institution can work alone. [211]

A me attitude, where the company's immediate financial balance is the only factor, limits network thinking. Not understanding how the actions of one node affect other nodes easily cripples whole segments of the network. ... Hierarchical thinking does not fit a network economy. In traditional organizations, rapid shifts can be made within the organization, which any resulting losses being offset by gains in other parts of the hierarchy. In a network economy each node must be profitable. [213]