*2.2. The Human Network of Physical Contacts*

#### 2.2.1. Brief Review of Small-World Networks

People establish many networks involving physical contact with each other through family relationships, friendships, sexual relationships, and many others. Networks have nodes (or vertices) and edges (connections). For example, consider the "handshaking" network in which each node is a person, and two persons are connected in this network if they had at least a handshaking, e.g., last year.

In a typical network established by people, each person connects to a tiny subset of another person included in that network. Therefore, most people have no direct connections (each one of us gave a handshake to just a few people last year).

In some networks involving people, if a given person connects to two others, these two persons are likely connected to each other, but each person can reach most people (through the network's connections) by a small number of connections. The last sentence sounds somewhat contradictory, but it is not–strikingly, many networks established by people are similar to this–the so-called small-world networks–and we will discuss how that is relevant to understanding microorganisms' spread.

We first consider a regular network (Figure 1, left panel) and then change it to make it a small-world network (Figure 1, middle panel). Consider, for example, the network of friendships. For clarification, let us assume that individuals in a population are organized in alphabetic order: A, B, C, ... , Y, Z, AA, AB, ... and that all individuals have precisely four friends. Frequently, if individual C is a friend of two individuals on his/her right, A and B, and the two individuals on his/her left, D and E, then probably B and D are friends of each other, and A and B or D and E.

Meanwhile, D is a friend of B, C (right), E, and F (left), so B and C are friends to each other, as well as E and F, and so on. Therefore, the clustering of these networks' nodes (people) is high. If all friendships were similar to this, the friendships' network would be regular. In such a network, if, for example, individual G has an exciting gossip, it takes nine steps to reach, say, individual X. These nine steps are the following: G first informs I, which would transmit the story to K, then M, O, Q, S, U, V, X.

**Figure 1.** Regular, small-world, and random networks.

Of course, some exceptions to the regular network of friendships may substantially impact information spread. For example, according to the rule described above, individual K may be a friend of I, J, and L. Nevertheless, in this new network version, the fourth K's friend is Z, not M. With this exception, the network is no longer a regular one. In this case, a gossip would take just four steps to progress from G to X (G => I => K => Z => X). With a few more changes in other individuals, such as the one we introduced in K, the network becomes a "small-world" network.

In small-world networks, the clustering of nodes (people) is high, but the path of friendships between any human being is short. Rumors may spread in regular networks, but the speed would be much higher in small-world networks [28].

What makes small-world networks so relevant to epidemiology? As mentioned above, if people organize themselves in small-world networks, the spread of information is fast because, although most people do not have direct contact with each other, most can be reached in a few steps. Suppose physical proximity or even contact is involved in these networks. In that case, microorganisms may quickly spread because the typical distance between two randomly chosen people (the network nodes) grows proportionally to its logarithm [28] instead of the number of people in the network as in regular networks. This difference is relevant because the logarithm function grows much slower than a linear function (Figure 2A). For example, when a given variable X increases from one to a billion (i.e., from 1 to 109), Log10X goes from zero to nine only. Therefore, in small-world networks, the path between two random persons is low, even if the network contains millions or billions of people.

**Figure 2.** (**A**): Linear and logarithmic functions. Suppose that x increases from 1 to 109, close to the world population size. Then, y increases from 1 to 109 in the case of the *y* = *x* linear function, whereas the *y* = *Log*10(*x*) function increases from zero to nine only. Note that both axes are at the logarithmic scale (base 10). (**B**): Exponential and power-law decay. The exponential function 2−<sup>t</sup> decreases much faster than the power-law function t<sup>−</sup>2. Note that the vertical axis is at the logarithmic scale (base 10) but not the horizontal axis.

Of course, people spontaneously build other contact networks. For example, people in our working place are not necessarily our friends, but we contact them daily. These networks, which usually involve physical proximity or sharing a working environment, may be relevant concerning the microorganisms' evolution or spread. For example, previous studies have shown that the shorter path lengths in small-world networks increase the effectiveness of natural selection while maintaining the fittest clones in bacterial populations because the probability of encounters between individuals is higher than in regular networks [29,30].

Liljeros et al. studied the web of human sexual connections among 2810 adult Swedish people and found that those connections defined a scale-free network [31,32]. In scale-free networks, the distribution of links in each node follows a power law [33]. In the case of the network of sexual contacts, *p*(*k*) describes the proportion of people with *k* sexual contacts in the previous year, *<sup>p</sup>*(*k*) <sup>≈</sup> *<sup>c</sup>*·*k*−*α*, where c and *<sup>α</sup>* are positive parameters. Therefore, all people in that Swedish database had at least one sexual partner in the previous year, but a lower fraction had two partners; an even lower fraction, some people had three sexual partners, and so on. Power-law distributions characteristically decreased slowly, so some individuals had 20 partners in the previous year. These few individuals are "hubs" of this network, which is relevant, for example, for sexually transmitted diseases [32]. In this study, the exponent *α* is slightly above three. The exponent *α* for women is higher than that for men, which means that the proportion of women with *k* sexual partners is lower than that of men; however, the statistical error of these estimations is higher than their difference [31,32,34]. The same authors have also shown that the cumulative distribution *<sup>P</sup>*(*k*) similarly follows a decreasing power-law distribution, *<sup>P</sup>*(<sup>≥</sup> *<sup>k</sup>*) <sup>≈</sup> *<sup>c</sup>*·*k*−*ρ*, where *ρ* = *α* − 1. The word "cumulative" in the previous sentence means that *P*(≥ *k*) quantifies the fraction of people with *k* or more contacts [31].
