Fluctuations in Network Dynamics

Most complex networks serve as conduits for various dynamical processes, ranging from mass transfer by chemical reactions in the cell to packet transfer on the Internet. We collected data on the time dependent activity of five natural and technological networks, finding that for each the coupling of the flux fluctuations with the total flux on individual nodes obeys a unique scaling law. We show that the observed scaling can explain the competition between the system’s internal collective dynamics and changes in the external environment, allowing us to predict the relevant scaling exponents.

Figure 1

(a) Time dependent traffic on three Internet routers of the Mid-Atlantic Crossroads network, whose activity is monitored by the Multi Router Traffic Grapher software (MRTG). The figure shows the number of bytes per second for each of the routers in 5 min intervals for a two day period. (b) Streamflow, measured in cubic feet per second, on three rivers in the U.S. river basin, based on data collected by the U.S. Geological Survey in $2001$. On the right of each plot we show the time average of the flux $⟨f⟩$ displayed as horizontal dotted lines superposed on the graphs, and the dispersion, $\sigma$, for each signal, indicating orders of magnitude differences in both flux and dispersion between nodes of the same network.

Figure 2

The relationship between fluctuations ($\sigma$) and the average flux ($⟨f⟩$) for the $\alpha =1/2$ systems. (a) Time resolved information for $374$ Internet routers of the Mid-Atlantic Crossroads, ABILENE network, MIT routers, UNAM routers, all Brazilian RNP backbones, and dozens of smaller routers on the Internet, covering for each node two days of activity with 5 min resolution. (b) The activity of the $462$ signal carriers of the 12-bit Simple12 microprocessor, recorded over $8862$ clock cycles.

Figure 3

The relationship between fluctuations ($\sigma$) and the average flux ($⟨f⟩$) for systems belonging to the $\alpha =1$ class. (a) Daily visitations on websites collected using the Nedstat web monitor. We analyzed daily traffic for a $30$ day period for $1000$ sites in USA (circles), Brazil (squares), and Japan (triangles). (b) The daily streamflow of $3945$ rivers on the U.S. river basin during the year of $2001$ is recorded by the U.S. Geological Survey. (c): Daily traffic on Colorado and Vermont highways representing the daily number of cars passing through observation points on $127$ highways from $1998$ to $2001$.

Figure 4

In model 1 on each “day” $t$ we release $W(t)=⟨W⟩+\xi (t)$ walkers on randomly selected nodes and allow them to perform $M={10}^3$ random diffusive steps, where $\xi (t)$ is a uniformly distributed random variable between $-\Delta W$ and $\Delta W$ and $⟨W⟩={10}^4$. (a) The figure shows the $\sigma (⟨f⟩)$ curves for $\Delta W=0,20,40,80,100,200,800,1000,4000,10000$ from top to bottom. (b) The dependence of the exponent $\alpha$ on $\Delta W$, obtained by fitting the ${\sigma }_i$ versus $⟨f_i⟩$ curves shown in (a). Note that while the figure shows a gradual transition, the transition in infinite systems should be sharp. (c) Average fluctuations $⟨{\sigma }_i⟩$, obtained by averaging ${\sigma }_i$ over all nodes $i$ in the system, shown in function of the amplitude of the external driving force $\Delta W$. While under $\Delta W\approx {10}^2$ the magnitude of $⟨{\sigma }_i⟩$ is independent of $\Delta W$, for large $\Delta W$ the fluctuations increase rapidly, indicating that the network dynamics is externally driven. (d)–(f) The same as in (a)–(c), but for model 2, where the diffusive dynamics was replaced by message passing. $W$ was again chosen from an uniform distribution of width $\Delta W$ and average $⟨W⟩={10}^4$. In all simulations we used a scale-free network [6] with $\gamma =3$ and ${10}^4$ nodes.