Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2007-02-28
Phys. Rev. Lett. 99, 188701 (2007)
Physics
Condensed Matter
Statistical Mechanics
11 pages, 3 figures, 1 table
Scientific paper
10.1103/PhysRevLett.99.188701
We study the stability of network communication after removal of $q=1-p$ links under the assumption that communication is effective only if the shortest path between nodes $i$ and $j$ after removal is shorter than $a\ell_{ij} (a\geq1)$ where $\ell_{ij}$ is the shortest path before removal. For a large class of networks, we find a new percolation transition at $\tilde{p}_c=(\kappa_o-1)^{(1-a)/a}$, where $\kappa_o\equiv < k^2>/< k>$ and $k$ is the node degree. Below $\tilde{p}_c$, only a fraction $N^{\delta}$ of the network nodes can communicate, where $\delta\equiv a(1-|\log p|/\log{(\kappa_o-1)}) < 1$, while above $\tilde{p}_c$, order $N$ nodes can communicate within the limited path length $a\ell_{ij}$. Our analytical results are supported by simulations on Erd\H{o}s-R\'{e}nyi and scale-free network models. We expect our results to influence the design of networks, routing algorithms, and immunization strategies, where short paths are most relevant.
Carmi Shai
Cohen Reuven
Havlin Shlomo
Lopez Eduardo
Parshani Roni
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