Solvable model for distribution networks on random graphs

We propose a simple model that captures the salient properties of distribution networks, and study the possible occurrence of blackouts, i.e., sudden failings of large portions of such networks. The model is defined on a random graph of finite connectivity. The nodes of the graph represent hubs of the network, while the edges of the graph represent the links of the distribution network. Both, the nodes and the edges carry dynamical two state variables representing the functioning or dysfunctional state of the node or link in question. We describe a dynamical process in which the breakdown of a link or node is triggered when the level of maintenance it receives falls below a given threshold. This form of dynamics can lead to situations of catastrophic breakdown, if levels of maintenance are themselves dependent on the functioning of the net, once maintenance levels locally fall below a critical threshold due to fluctuations. We formulate conditions under which such systems can be analyzed in terms of thermodynamic equilibrium techniques, and under these conditions derive a phase diagram characterizing the collective behavior of the system, given its model parameters. The phase diagram is confirmed qualitatively and quantitatively by simulations on explicit realizations of the graph, thus confirming the validity of our approach.

Figure 1

A section of the phase diagram in the $\left(a,\mu \right)$ plane, with $b=d=0$ for three different average connectivities: $C=4$ (full lines), $C=6$ (dashed lines), and $C=8$ (dotted lines). For each, the top line is the upper spinodal [marking the boundary between the (CO) and the (O) region], the middle line is the equilibrium transition, and the bottom line is the lower spinodal [marking the boundary between the (CNO) and the (N) region]. Note that for average connectivities $C=6$ and 8, the transition and the lower spinodal are so close that they cannot be distinguished on the scale of this plot, thus reducing the (CN) region to a very narrow strip.

Figure 2

A section of the phase diagram in the $\left(a,\mu \right)$ plane, with $d=0$, $C=4$ for three different values of $b$: $b=3$ (full lines), $b=0$ (dashed lines), and $b=-3$ (dotted lines). For each, the top line is the upper spinodal, the middle line is the transition, and the bottom line is the lower spinodal.

Figure 3

A section of the phase diagram in the $\left(a,\mu \right)$ plane, with $b=0$, $C=4$ for three different values of $d$: $d=3$ (full lines), $d=0$ (dashed lines), and $d=-3$ (dotted lines). For each, the top line is the upper spinodal, the middle line is the transition, and the bottom line is the lower spinodal.

Figure 4

The fraction of nodes that are 1 $\left(m_n\right)$, as a function of $a$ and $\mu$ for $C=4$, $b=d=0$, both starting from the (N) phase and the (O) phase.

Figure 5

Monte Carlo simulation of a system of ${10}^5$ nodes, with parameters $C=4$, $⟨\mu ⟩=-8$, $⟨b⟩=0$, and $⟨d⟩=0$, exhibiting hysteresis of $m_n$ as a function of $⟨a⟩$ across the transition. Connecting solid lines are for uniform parameters $\left(\sigma =0\right)$, dotted lines are for nonuniform parameters with $\sigma =0.5$. The symbols “$\times$” indicate that we started from the (O) solution, while “$+$”indicates we started from the (N) solution. Note that the fluctuations of $m_n$ over the measuring sweeps are of the order of the line widths. We used ${10}^2$ equilibration and measuring sweeps for each value of $⟨a⟩$.

Figure 6

Simulation of a distribution network of 75 nodes, with constant parameters $C=4$, $⟨\mu ⟩=-8$, $⟨b⟩=0$, $⟨d⟩=0$, and $\sigma =0$. Plotted is $m_n$ as a function of time (measured in Monte Carlo steps per degree of freedom). Starting from the metastable (O) phase, the system collapses after about ${10}^5$ sweeps to the globally stable (N) phase. Note that the large fluctuations of $m_n$ are due to the small system size.