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Linked - Book Review

1 December 2007
Dr. Moria Levy

It would be the epitome of popular science if one encapsulates the greatness of the book "Links," originally titled "Linked," in a single phrase. In a positive sense, this work by a physicist delves into a scientific subject rooted in mathematics and reaches across diverse scientific realms. The author articulates these concepts in clear, fluent language, presenting simple, everyday examples and more intricate ones. The use of language facilitates reader comprehension, making it easier for us to grasp the content. The book imparts knowledge about how various aspects of our lives, seemingly disparate, adhere to a familiar pattern – a pattern of networks.

Rather than commencing with a description of the model developed by the author and colleagues, the book unfolds gradually in both time and complexity, involving the reader as a partner in the process. Engaging in knowledge management, the reader naturally poses pertinent questions, prioritizing and sequencing information. This article, summarizing key points from the book, does not seek to provide a comprehensive overview but instead describes the model, mentions applications, and raises questions within our realm, the world of knowledge management, regarding its application in our context.

Published in 2002 (with an updated edition in 2004), the book is akin to a set of finely crafted Lego blocks. It offers numerous building blocks and answers. Nevertheless, it also allows ample space for reflection and imagination on the reader's part.

The book covers the following topics:

  • Random world

  • A world of clusters

  • A world of hubs

  • A world of individual winners

  • Degrees of separation

  • Stability and instability

  • Growth and proliferation of networks

  • Hubs as distribution factors

  • Applications

  • The world of knowledge management

  • Items of knowledge

Random world

The realm of networks is a relatively recent domain, with its roots traced back to Euler, one of the greatest mathematicians, and his 1736 paper known as Kniseberg's Bridge Essay. Euler laid the groundwork for graphs, a combination of nodes and links, establishing the foundation for analyzing graphs and their properties. Jumping ahead two centuries to 1959, young mathematicians Erdős and Reney developed the theory of random networks, initially aimed at revealing the mathematical beauty behind various phenomena, from natural creatures to human actions and social behaviors.

The theory of random networks is based on the concept that in a network, nearly every node will have roughly the same number of links as others. This holds across different scenarios, whether examining the average number of connections people have, roads connected to other roads, or entities linked to suppliers and customers. The likelihood of encountering a significantly unusual node (completely isolated or with a much higher number of links than its peers) is exceedingly low. Substantial deviations from the average are rare, rendering such a network notably resilient. Even in the event of damage to a node, whether random or non-random, alternative paths exist to maintain the network's continuity.

In such a network, the number of links to each node can be described using a Gaussian distribution, also known as a bell distribution, due to its shape. Many nodes have the same number of links as the average, gradually decreasing until they reach the reset point (as there are no unusual deviations). This distribution may bring to mind university grade distributions when professors applied bonuses (grade corrections) to ensure a natural distribution.

Here (on the right) is a sketch of a random grid and a diagram of a bell distribution (on the left):

While the depiction might evoke memories of forgotten mathematical concepts, the book's purpose isn't to delve into mathematical intricacies. Instead, its essence lies in teaching us about the behavior of people and networks, including social networks, and how these networks exhibit similarities with numerous others. The structure of random networks and this distribution type prove crucial, as explored later in the book.

A world of clusters

Clusters, or more straightforwardly, grouped groups, represent subcollections within a network interconnected by multiple connections. While each partner's links to the broader network are sparse and random, connections are established with almost everyone within clusters. This natural phenomenon mirrors our society, where human beings endeavor to create intimacy within various circles or clusters: work circles, circles of friends, and family circles. In these clusters, individuals are closely connected to almost all other members.

Clusters are not exclusive to social networks alone; they characterize various networks, including the neural network in our brain, ecosystems, web systems, economic systems, and more. The concept of networks as clusters, introduced in a model published in a 1998 paper by Watts and Strogats, piqued significant interest among scientists. One intriguing feature of these networks is that, despite the close ties within clusters, often higher significance is attached to weak connections outside the cluster.

A study conducted in the United States revealed that in job searching, individuals rely more on distant friends (weak ties) than on close friends within the cluster. The rationale is that information about job opportunities within the cluster is likely already known to us through our close friends. The weak connections, acting like bridges, open up new opportunities and provide us with influence beyond our cluster. These weak connections transform our world into a "small" world, allowing us to reach parts seemingly far from our current network node. With only a few weak connections, large networks converge, facilitating quick access from almost any node to any other node. While somewhat surprising, this feature is crucial as it ensures continued security within the clusters without the risk of disconnection from other network parts.

Here's an illustration of a cluster-based world:

A world of hubs

From chapter to chapter, we discover that the world of networks is not as randomly structured as initially believed. It proves to be more intricate, and understanding its organization sheds light on its properties. While the concept of random networks seems accurate at first glance, a closer examination of their internal structure unveils that networks comprise clusters and subclusters.

An unexpected revelation surfaces within these clusters: contrary to expectations, the clusters encompass diverse nodes. While most nodes within a cluster share similarities in their internal and external links, only a few nodes in the network can be deemed "ultra-popular." These nodes, reminiscent of hubs, wield dominance over the entire network. For example, in the online realm, Amazon and Google may act as hubs; in social networks, they could be widely recognized individuals, and this pattern extends across various networks, from cellular structures to chemical reactions and phone call networks.

Despite challenging the expectations set by the original models of random networks and networks of clusters, these hubs not only thrive but also play a significant role in multiple domains. The 80-20 rule, or Pareto Principle, applies here: 80% of links belong to 20% of network nodes (hubs). In mathematical terms, this property is encapsulated in a law known as the law of possession. Unlike the bell curve used to describe the distribution of links to different nodes, these networks are characterized by a strong distribution curve, referred to as scaleless networks. The network structure, when depicted, resembles a map of major airports in the US, with a few significant hubs connecting to many others.

Although the significance of hubs might not be immediately apparent, there's profound meaning embedded in the laws of possession. These laws transform the world from disorder to order, like transitions from liquid to solid or disorder to magnetism. While we can identify hubs at this stage, the mechanisms that give rise to them remain unclear. The reasons for their emergence and whether it implies a constant transition from disorder to order, especially in the realm of the web, still need to be understood. The subsequent exploration of the world of hubs delves into unique features such as battles between nodes, high stability, and simultaneous risks of instability, as discussed further in the book.

Here is an illustration of a scaleless grid and a strong distribution curve:

A world of individual winners

Not all networks adhere to the discussed randomness. A few select networks exist where the previously described topology (a network lacking a scale) is absent. A distinct node emerges in these networks, eventually taking control of the entire network and evolving into a singular hub, connecting everyone to it. A notable example from the software world is Microsoft, which achieved such dominance. In the realm of research, this phenomenon is associated with the Bose-Einstein condensation principle (details of which are beyond the scope of this summary).

This phenomenon disrupts the typical competitiveness found in ordinary networks, be they random or lacking a scale. It often occurs in a network without a scale when one node manages to attain such dominance that all new links converge toward it. This configuration, known as a "star" or "winner takes everything," can be depicted using the following diagram:

Degrees of separation

Many of us are acquainted with the Six Degrees of Separation rule, which posits that any pair of people on the planet can be linked through up to six intermediaries (acquaintances) on average. This phenomenon is most notable in a world with six billion people. It originated from a story by Karniti, a mathematician, titled "Chains," from the early 20th century and was later popularized in the play of the same name in 1991. Despite the many nodes, humanity is revealed as a very dense network.

A study conducted by the author of the book and his collaborators indicated that the number of links required on the web (based on an estimate at the end of 1998 of about 800 million nodes) could be expected to be 18.59 degrees of separation. In other words, each document is an average of 19 clicks away from any other document. Reflecting on and understanding the world of networks, which now includes not only random networks but also clusters and, more importantly, hubs, it becomes clear why such a feature could emerge.

Nevertheless, the phenomenon is fascinating and raises high expectations for connectivity. What is disappointing is that finding a shortened route to our destinations is very challenging. Despite the network's extreme density, our ability to navigate and not find what we want surpasses our chances of reaching a safe shore. Numerous long routes exist, and we still lack an algorithm to ensure we are progressing, getting closer, and not straying far from our goal. Note: The exact number is 5.5, and 6 is an approximate figure.

Stability and instability

A scaleless network imparts a sense of stability. Random damage to one node or another is less likely to impair network connectivity. Even if a hub is targeted, the likelihood of bringing down the network by damaging its connectivity is very low. Systems, especially natural ones, exhibit high survivability. Despite internal failures affecting their behavior, they usually fulfill their purpose, even under extremely high error rates. The unexpected resilience against malfunctions is a distinctive feature of scaleless networks that is absent in random networks.

Unfortunately, scaleless networks are not invulnerable. In 1996, the entire power grid from the Rocky Mountain Range to the Pacific Ocean experienced a shutdown. Similarly, biological systems collapse due to damage to specific proteins and more. It is crucial to understand how long it takes for the network to break down once nodes are removed from it. Although removing more nodes isolates more clusters, decades of research have revealed that dismantling the network is not gradual. Once the number of removed nodes reaches a critical point, the system abruptly breaks down into small, unconnected islands, resulting in a system crash. A vital failure threshold exists below which the system is relatively unaffected, but above it, the network falls apart.

A random collapse is a rare event, but the vulnerability of networks to attacks arises from the characteristics of hubs. Understanding the network and damaging multiple hubs almost immediately brings down the entire network. This vulnerability was exploited in events like the September 11 attacks and forms the basis of many modern terrorist networks, focusing on hitting social, political, or infrastructure hubs. Resisting such an attack is challenging, as a cascade of failures is triggered, causing the entire network to crumble before our eyes. Unfortunately, the same vulnerability applies to body systems, the electric grid, and nearly every other critical network. Topological stability does not provide resilience against attacks; quite the opposite.

Note: The ability to traverse from any node to any node via linked intermediate nodes.

Growth and proliferation of networks

In contrast to random networks, the key to developing scaleless networks lies in closely examining how networks grow and expand. While networks may appear static at a particular moment, most scaleless networks are in a constant state of growth. The developmental model, articulated by Barbashi and colleagues, is quite simple and is defined by two rules:

  1. Growth: A new node is added to the network during any given period. This step underscores the idea that networks form through the sequential addition of nodes.

  2. Preferred Connection: Each new node is assumed to connect to existing nodes through two links. The probability of choosing a particular node is proportional to the number of links the selected node already possesses. In other words, if there are two different nodes in front of the new node, one with 'n' links and the other with '3n' links, there is a threefold likelihood that the new node will connect to the latter. This phenomenon is known as "the rich get even richer," explaining phenomena such as the advantage of veterans, adherence to WEB2 principles favoring early adopters, and more.

The combination of these two laws effectively explains the development of scaleless networks. Understanding the developmental pattern has led to a prosperous and cohesive network growth and development theory, evolving over the past decade. The significance of this understanding lies in its ability to accurately predict scaling exponents and dynamics of networks, contributing to an understanding of the complex architecture of network structures. The matching parameter determines whether a network remains random, develops without a scale, or evolves into a single winner. The matching distribution illustrates the similarity between nodes. The more similar nodes are to each other, the more uniformly new nodes will be drawn to connect, leading to a bell-shaped distribution characteristic of random networks. Conversely, more significant variation in matching distributions (higher success rates for players, more professionalism in companies, more significance for proteins) increases the likelihood of creating a scaleless network with a power-law distribution. In extreme cases, where a new node with significantly higher compatibility emerges, there's a chance of transitioning into a "winner takes everything" network.

Hubs as distribution factors

Hubs transcend being mere links between nodes; they evolve into what the marketing world recognizes as "opinion leaders." Even if hubs lack innovativeness or inventiveness, they tend to be exposed to innovations before most other nodes. Due to their extensive connections, innovations reach hubs before any random node. A hub's adoption of a product or opinion and its discourse on a phenomenon or idea plays a crucial role in successfully disseminating that concept throughout the entire network.

Numerous studies have scrutinized social and professional networks more closely in recent years. While sociologists and marketing experts have long acknowledged this phenomenon, its consistency and high level of influence are now better understood. Despite discussing favorable distribution, it's essential to recognize that alongside it, there exists a less favorable distribution.

In virus models (whether for diseases like AIDS or computer viruses), the threshold model was traditionally emphasized: the virus would only continue to spread if its strength exceeded a certain threshold. This holds in a random network. However, in a network without a scale, where hubs play a critical role, viruses spread even without surpassing a threshold. As hubs are connected to many nodes, their chances of infection are not random but highly high. Infecting hubs effectively results in the entire network being infected. The threshold model is irrelevant in such networks, and viruses persist in spreading even when their levels are below the minimum threshold.

This understanding leads to a new, if not sociological, policy: diverting resources to care for hubs instead of uniformly distributing resources. While adopting such an approach may not always be easy, its effectiveness becomes evident when the infection and distribution model are clear.


In this chapter, we deviate slightly from the original structure. The book extensively explores applications across various domains and needs to be more extensive to list them comprehensively. To give readers a glimpse of the vast array of scaleless networks, we will mention some of the worlds covered in the book, including examples from previous chapters and some that haven't been discussed in detail. For a deeper understanding of their behavior and the impact of scaleless networks on our daily lives, readers are encouraged to refer to the book itself. Scaleless networks covered include:

  • The Internet (the network of computers interconnected via the Internet)

  • Electric grid

  • Airport Network

  • The Web (the pages and documents linked over the Internet)

  • Cell grid

  • Metabolism Map

  • Neurons in our brains

  • The economic network of companies

  • Network of Directors of Companies

  • Network of financial institutions

  • And the list goes on and on.

The world of knowledge management

I want to note that this chapter is not part of the original book; it includes my insights as the article's author regarding the discussed content in the context of our world—the world of knowledge management.

Knowledge management can be broadly divided into two sub-worlds: the management of the knowledgeable and the management of knowledge items. Knowledgeable individuals are integral to communities and social networks. Barbashi posits that communities exhibit behavior akin to networks lacking scale: at their core are several individuals significantly connected to more people than anyone else. As per the examinations, engaging with these individuals triggers an expansion effect. Persuasive activities for utilization and marketing should be directed toward them. Who comprises this group? Are they content experts? Possibly, but not necessarily. While those with deep wisdom and extensive knowledge may have many connections (as seen in Leonard and Swap's book "Deep Smarts"[1]), not all content experts possess profound intelligence. We label the well-connected nodes of knowledge as "knowledge nodes" (refer to the term "knowledge node" on the KMROM website glossary [2]). Unsurprisingly, these nodes of expertise align with the hubs Barbashi discusses.

According to Barbashi's model, this holds several implications beyond what we have previously attributed to these individuals:

  • Invest Change Management efforts in these people.

  • Construct expert maps with them.

Note: Expert maps are a much more effective and central tool in knowledge management solutions than previously thought.

Items of knowledge

Knowledge items encompass the elements we manage through sharing and retention tools, such as websites, communities, and insight repositories. Several immediate questions arise within this realm: Are knowledge items a network? If so, what is the structure of this network? What constitutes the nodes and links between different pieces of knowledge? Is it a random network or a scaleless network characterized by hubs?

Drawing on research analyses from social and professional networks, these networks appear to lack scale. Consequently, effective knowledge management might be simplified by identifying "hub" items in each field one wishes to manage. Managing these "hub" items could yield a significant return, aligning with the Pareto Law (20:80), as in most cases, the goal is not to manage all knowledge but only the most valuable. This approach resonates with the modus operandi outlined in Collison and Purcell's book, "Learning to Fly," particularly in the chapter on knowledge assets. However, even there, the focus isn't on "hubs." This serves as the brief core of knowledge from which to commence.

At this stage, it's challenging to definitively assert that the world of knowledge operates as a scaleless network of hubs. While it likely exists, recognition remains elusive. Future developments might unveil a formula or condition for when the structure of knowledge relationships transitions into a scaleless structure. The network hasn't reached a sufficient growth stage, and time will reveal our direction.

[1] "Deep Smarts," Harvard Business School, 2005


[3] "Learning to Fly," Capstone Publishing, 2001, 2004

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