Figure 1

Evolution of eight agents on a $5\times 5$ lattice for $t$ from 0 to 3. For each $t$ the lattice is shown on the top and the Information Tree is shown on the bottom. Informed agents are black circles; unaware agents are white circles. A gray circle of radius 1 is drawn around every informed agent to represent its action (an agent is in contact with another if it falls within this circle). $t=0$: the only informed agent is the Information Source that carries information 1, so $n(0,0)=1$ and $n(0,l)=0$ for $l>0$. $t=1$: agent 1 passes information to agent 2; now $n(1,0)=1$, $n(1,1)=1$. $t=2$: agent 1 passes information to agent 3 and agent 2 passes information to agent 4; $n(2,0)=1$, $n(2,1)=2$, $n(3,1)=1$. $t=3$: agent 2 passes information to agent 5; agent 4 passes information to agents 6, 7, 8. Notice that agent 6 is in contact with both 3 and 4; it chooses randomly to get information from 4 (the same for agent 8). Now all agents have been informed: for this simulation the Population-Awareness Time is $\tau =3$. The final information is $\mathcal{I}\left(\tau \right)={\sum }_{l=0}^{\tau }n(l,\tau )z^l=1+2z+2z^2+3z^3$.Reuse & Permissions

Figure 2

(Color online) Evolution of $n\left(t\right)$ for a population of $N=32$ agents on six different lattices of size $L=2^m$, $m=4,\dots ,9$. Full circles denote the Population-Awareness Times, empty circles the Outbreak Times.Reuse & Permissions

Figure 3

(Color online) Dependence of the Population-Awareness Time $\tau$ on the number of agents $N$ and the lattice size $L$. Top: Log-log scale plot of $\tau$ versus $N$; different lattice-size values are shown with different symbols and colors. For sufficiently small densities $(\rho ⩽1)$, straight lines represent the best fit according to Eq. (4). Bottom: Log-log scale plot of $\tau$ versus $L$; different values of the number of agents are shown with different symbols and colors. Provided that the density $\rho$ is not large $(\rho ⩽1)$, data points lay on the curves given by Eq. (4), which represent the best fit. Error on data points is $<2\%$.Reuse & Permissions

Figure 4

(Color online) Time evolution of level populations $n(l,t)$ for $N=32$, $L=512$.Reuse & Permissions

Figure 5

(Color online) Top: typical population distribution on levels at $t=\tau$ for a low-density system $(N=1024,L=4096)$. Squares are experimental results; the line is the result of data fitting according to Eq. (6). Bottom: population distribution on levels at $t=\tau$ for systems with $N=1024$ and $L$ between $2^4$ and $2^{12}$ (the lines are guides to the eye): the behavior of the distribution is nonmonotonic with respect to $L$. By increasing $L$ from small values, the curves first shift to the right and flatten $(L=16,24,32)$. The rightmost, extremal curve corresponds to $L=64$. Then, by increasing $L$ the curves shift back to the left and sharpen $(L=128,512,4096)$.Reuse & Permissions

Figure 6

(Color online) Semilog scale plot of final degree of information per agent ${\mathcal{I}}_{av}\left(z\right)=\mathcal{I}\left(z\right)∕N$ vs lattice size $L$. Several values of $N$ are shown with different symbols and colors (lines are guides to the eye), while the decay constant is fixed at $z=0.9$. Notice the occurrence of minima at $\tilde{L},\tilde{N}$, and that $\tilde{L}$ is monotonically increasing with respect to $\tilde{N}$. Error on data points is $<1.5\%$.Reuse & Permissions

Figure 7

(Color online) Semilog scale plot of final degree of information per agent ${\mathcal{I}}_{av}\left(z\right)$ as a function of the lattice size $L$, when $N=2^9$ (lines are guides to the eye). Four different values of decay constant $z$ are considered, as shown by the legend.Reuse & Permissions

Figure 8

(Color online) Final $(t=\tau )$ degree of information per agent ${\mathcal{I}}_{av}\left(z\right)$ versus the decay constant $z$. The size of the lattice is fixed as $L=2^4$, while several values of $N$ are considered and represented in different colors and symbols. The curve depicted is the best fit when $N=2^{10}$ $(\rho >1)$, according to Eq. (8). Notice the existence of the fixed points $z=0$, ${\mathcal{I}}_{av}\left(z\right)=\frac{1}{N}$, and $z=1,{\mathcal{I}}_{av}\left(z\right)=1.$Reuse & Permissions

Figure 9

(Color online) Evolution of the system in the high-density approximation for a lattice with $L=12$ and a Source starting in the site with coordinates $(7,7)$. Left: the wave front of information on the lattice at times $t=1$, 2, 3, 4 has a square shape. Agents in the interior of the square are informed, agents on the exterior are unaware. Times correspond to levels: agents between the front at time 3 and that at time 4 belong to level 4, and so on. Right: final distribution of agents on levels; broken lines highlight the triangular shape of the distribution.Reuse & Permissions

Figure 10

Snapshots of four systems with $N=1024$ and $L=16$, 32, 64, 512, all at an instant near to the half-filling time. Only informed agents are shown; they are represented as circles of radius 1. Notice that the high-density picture of a connected set of informed agents holds up to $L=\tilde{L}=64$. For $L⩾64$, the picture breaks down and the system is better described by a low-density approximation $(L=512)$.Reuse & Permissions