4
Chapter 1. Introduction
are low.
If the relationship is one with real activity, and real associated costs for maintaining it, then the Dunbar Number seems to be a quite hard limit.
• Many properties of social networks exhibit power laws. Histograms of how
frequently some property is present, plotted in descending order of frequency,
create curves that drop off extremely quickly. This drop-off is so steep that
plotting both axes on logarithmic scales produces almost straight lines (whose
slopes characterize global network properties).
The first, and most important, implication of the presence of power laws is that
average properties are almost always meaningless. For example, we expect
the number of connections individuals have to vary widely, with relatively
few having many connections and many having relatively few connections.
The average number of relationships within a social network does not provide
an accurate picture of how an arbitrary individual is connected because the
distribution is so extremely skewed.
For example, nodes such as national leaders may have very large degrees indeed. Individuals such as Queen Elizabeth or the Dalai Lama are plausibly
within three steps of most of the world’s population. Their neighborhood of
diameter less than or equal to 3 is roughly 7 billion. A very outgoing person
who has 125 immediate neighbors still will not have (125) 2 connections at
distance two, or (125) 3 connections at distance three, because many of these
neighbors, and neighbors of neighbors, will know one another. Hence their
neighborhood of diameter less than or equal to 3 might only be a few thousands. A socially isolated person might only have a neighborhood of diameter
less than or equal to 3 of size 5 3 = 125. Thus neighborhood sizes can easily
vary by a factor of 10 7 .
• There is structure within the set of connections that each individual has. Typically, each individual has really close connections to k others, slightly less
close connections to k 2 others, even less close connections to k 3 others, and
perhaps tenuous connections to k 4 others. k typically has a value between 3
and 4, so the sum of the number of individuals in all of these layers agrees
well with the total given by the Dunbar Number [122]. For example, the first
layer for most people consists of family; the second layer of close friends; and
so on. (It also seems plausible that there is a further layer of k 5 connections
that reflects weaker forms of acquaintance — for example, many social media
platform users have roughly 400 “friends”.)
• The degree of an individual tends to be similar to the degrees of the others to
whom that individual is connected. This property is called assortativity. In
other words, if a person has many friends, then his or her friends will also
tend to be people with many friends (and vice versa: if they have only a few
friends, these friends will also tend to have only a few friends). This is very
different from the networks connecting computers, where a node with high
Chapter 1. Introduction
are low.
If the relationship is one with real activity, and real associated costs for maintaining it, then the Dunbar Number seems to be a quite hard limit.
• Many properties of social networks exhibit power laws. Histograms of how
frequently some property is present, plotted in descending order of frequency,
create curves that drop off extremely quickly. This drop-off is so steep that
plotting both axes on logarithmic scales produces almost straight lines (whose
slopes characterize global network properties).
The first, and most important, implication of the presence of power laws is that
average properties are almost always meaningless. For example, we expect
the number of connections individuals have to vary widely, with relatively
few having many connections and many having relatively few connections.
The average number of relationships within a social network does not provide
an accurate picture of how an arbitrary individual is connected because the
distribution is so extremely skewed.
For example, nodes such as national leaders may have very large degrees indeed. Individuals such as Queen Elizabeth or the Dalai Lama are plausibly
within three steps of most of the world’s population. Their neighborhood of
diameter less than or equal to 3 is roughly 7 billion. A very outgoing person
who has 125 immediate neighbors still will not have (125) 2 connections at
distance two, or (125) 3 connections at distance three, because many of these
neighbors, and neighbors of neighbors, will know one another. Hence their
neighborhood of diameter less than or equal to 3 might only be a few thousands. A socially isolated person might only have a neighborhood of diameter
less than or equal to 3 of size 5 3 = 125. Thus neighborhood sizes can easily
vary by a factor of 10 7 .
• There is structure within the set of connections that each individual has. Typically, each individual has really close connections to k others, slightly less
close connections to k 2 others, even less close connections to k 3 others, and
perhaps tenuous connections to k 4 others. k typically has a value between 3
and 4, so the sum of the number of individuals in all of these layers agrees
well with the total given by the Dunbar Number [122]. For example, the first
layer for most people consists of family; the second layer of close friends; and
so on. (It also seems plausible that there is a further layer of k 5 connections
that reflects weaker forms of acquaintance — for example, many social media
platform users have roughly 400 “friends”.)
• The degree of an individual tends to be similar to the degrees of the others to
whom that individual is connected. This property is called assortativity. In
other words, if a person has many friends, then his or her friends will also
tend to be people with many friends (and vice versa: if they have only a few
friends, these friends will also tend to have only a few friends). This is very
different from the networks connecting computers, where a node with high