Current redistribution in resistor networks: Fat-tail statistics in regular and small-world networks

The redistribution of electrical currents in resistor networks after single-bond failures is analyzed in terms of current-redistribution factors that are shown to depend only on the topology of the network and on the values of the bond resistances. We investigate the properties of these current-redistribution factors for regular network topologies (e.g., $d$-dimensional hypercubic lattices) as well as for small-world networks. In particular, we find that the statistics of the current redistribution factors exhibits a fat-tail behavior, which reflects the long-range nature of the current redistribution as determined by Kirchhoff's circuit laws.

Figure 1

Definition of current-redistribution factors ${\mathrm{\Delta }}_{\parallel }(x,y)$ and ${\mathrm{\Delta }}_{\perp }(x,y)$ in a 2D square lattice.

Figure 2

Infinite 2D square lattice: Comparison of exact values for $|{\mathrm{\Delta }}_{\parallel }|(0,y)$ with their asymptotic approximations, $\delta |{\mathrm{\Delta }}_{\parallel }|=\left[|{\mathrm{\Delta }}_{\parallel }{|}^{\mathrm{exact}}-{\left|{\mathrm{\Delta }}_{\parallel }\right|}^{\mathrm{asymp}}\right]/{\left|{\mathrm{\Delta }}_{\parallel }\right|}^{\mathrm{exact}}$.

Figure 3

Current-redistribution factors in a 2D square lattice consisting of $96\times 96$ sites and assuming periodic boundary conditions. The failing bond is oriented horizontally in the center of the graph, i.e., connects sites $(0,0)$ and $(1,0)$. The left (right) panel shows the current-redistribution factors in bonds parallel, ${\mathrm{\Delta }}_{\parallel }$ (perpendicular, ${\mathrm{\Delta }}_{\perp }$), to the failing bond. Regions of positive and negative $\mathrm{\Delta }$ are indicated.

Figure 4

Cumulative distribution function of the size-scaled magnitude of the current-redistribution factors $\widehat{\mathrm{\Delta }}=N\left|\mathrm{\Delta }\right|$ for a 2D square lattice of $96\times 96$ sites assuming periodic boundary conditions. Dotted (red) line: result for all bonds. Dashed (blue) line: only bonds within a distance $d\le 24$ of the failing bond. The analytical result (3.16) for an infinitely large system within a continuum approximation is indicated by the solid line. The inset shows the tail of the distribution function for large $\widehat{\mathrm{\Delta }}$. Crosses (red): all bonds. Squares (blue): only bonds within a distance $d\le 24$ of the failing bond.

Figure 5

Cumulative distribution function of the distance-scaled magnitude of the current-redistribution factors $\tilde{\mathrm{\Delta }}=\pi d^2\left|\mathrm{\Delta }\right|$ for a 2D square lattice of $96\times 96$ sites assuming periodic boundary conditions. The meaning of the symbols is as in Fig. 4 and the solid line denotes the analytical result (3.20).

Figure 6

Semi-infinite ladder consisting of bonds with unit conductances along the ladder as well as for the leftmost spoke (solid line), which is assumed to fail, and equal conductances $g$ of the other spokes (dashed line).

Figure 7

Ring with one long-range connection.

Figure 8

Behavior of (a) $\langle \langle \tilde{\delta }\rangle \rangle$ [Eq. (5.4)] and (b) $ln\langle \langle \left|\mathrm{\Delta }\right|\rangle \rangle$ [Eq. (5.2)] vs $\delta$.

Figure 9

Behavior of $s(g,N)$, the exponent that characterizes the decrease of $\left|\mathrm{\Delta }\right(d\left)\right|$ in a fully connected ring of size $N$ with long-range conductances $g$. Comparison is made with the corresponding exponent of a one-dimensional ladder with spoke conductances $gN/2$; see Eq. (5.11).

Figure 10

Averaged $\left|\mathrm{\Delta }\right|$ as a function of the scaled ring distance $pd$ for rings of size $N=4000$ (red, open symbols) and $10000$ (blue, closed symbols), and for different values of the probability parameter $p$. The numerical results represent averages over both the location of the failing bond and 25 realizations of the small-world network. The lines serve as a guide to the eye. The arrows indicate the location of the critical ring distances $d_0(p,N)$ [Eq. (5.14)].

Figure 11

Average $\langle \langle \left|\mathrm{\Delta }\right|\rangle \rangle$ of the current-redistribution factors $|{\mathrm{\Delta }}_{ij,mn}|$ for rings of different size $N$ and for different probabilities $p$ of adding long-range connections. The numerical results represent averages over both the location of the failing bond and 25 realizations of the small-world network. The line marked “Lorentz” corresponds to the approximation of Eq. (5.20) for $\langle \left|\mathrm{\Delta }\right|\rangle$ vs $pN$.

Figure 12

Cumulative distribution function of the scaled current-redistribution factors, $Np\langle \left|\mathrm{\Delta }\right|\rangle$, for a ring with $N=10000$ sizes and different $p$ values for two different realizations of the random long-range links (solid and dashed lines). The dotted curve denoted by “Lorentz” corresponds to Eq. (5.19).