The topology of a network directly influences the diffusion of information over the network. Imagine a scenario in which nodes receive some kind of information (an idea, a rumor, a fad, a virus, and the like) and might decide to spread it over the network through their adjacent nodes. Is this information going to become widely popular over the network?
Suppose each node that received the information communicates it with independent probability $r$ to its immediate neighbours. If a node has degree $k$, then it can pass the information to $r (k-1)$ neighbours on average, not counting the one from whom the information is arrived. Taking a weighted average over all nodes, the average number of nodes an individual node passes the information is:
$\displaystyle{R_0 = \frac{\sum_i k_i r (k_i-1)}{\sum_i k_i} = r \frac{\langle k^2 \rangle - \langle k \rangle}{\langle k \rangle}}$
where $k_i$ is the degree of node $i$, $\langle k \rangle = (1/n) \sum_i k_{i}$ is the mean degree, and $\langle k^2 \rangle = (1/n) \sum_i k_{i}^{2}$ is the mean-square degree.
The quadratic degree at numerator of the above formula has an intuitive explanation: the $i$th node can hear an information from $k_i$ other nodes and spread it with probability $r$ to $k_i$ neighbours minus the information source node.
If $R_0 > 1$, then the number of nodes hearing about the information grows exponentially and the idea is going to become quickly popular over the network community. Otherwise, the idea will not make the headlines. Notice that $R_0 > 1$ if and only if the communication probability $r$ is larger than a tipping point:
$\displaystyle{r > \frac{\langle k \rangle}{\langle k^2 \rangle - \langle k \rangle}}$
Notice that for networks with degree distribution following a power law, or more generally for those having a fat right tail, the value $\langle k^2 \rangle$ is big compared to $\langle k \rangle$, and hence the probability $r$ that each individual spreads the information need not be large for it to reach the whole community.
tip = function(g) {
d = degree(g);
k = mean(d);
k2 = mean(d * d);
return(k / (k2 - k))
}
g = barabasi.game(1000, m = 2, directed=FALSE, out.pref = TRUE)
tip(g)
[1] 0.1183229
tip(erdos.renyi.game(vcount(g), ecount(g), type = "gnm"))
[1] 0.2509420
Putting it another way, hubs -- nodes with a disproportionately large number of contacts -- play a dual role in information diffusion over scale-free networks: on the one hand, they quickly harvest information, on the other hand, they effectively spread it. This is why even mildly contagious (computer and biological) viruses widely spread and persist over social networks, defying the predictions of the classical epidemic models.
The topology of a network also influences its resilience to failures and targeted attacks. A scale-free network is glued together by hubs. Hubs are rare compared to ordinary nodes. Since failures affect random nodes in a network, the probability that a hub is damaged by a failure is relatively low. The price of this resilience, however, is the vulnerability to malicious attacks. These aim to dismantle the hubs of the web, not the ordinary nodes. An attack to few of this hubs is in fact sufficient to break the network into disconnected small pieces. Hence, scale-free networks are robust under failures but fragile under attacks. On the other hand, on random networks there is no difference between failures and attacks, since each node has approximately the same number of connections.