Trapping in complex networks

We investigate the trapping problem in Erd˝ os-Renyi (ER) and scale-free (SF) networks. We calculate the evolution of the particle density ρ(t) of random walkers in the presence of one or multiple traps with concentration c. We show using theory and simulations that in ER networks, while for short times ρ(t) ∝ exp(−Act), for longer times ρ(t) exhibits a more complex behavior, with explicit dependence on both the number of traps and the size of the network. In SF networks we reveal the significant impact of the trap's location: ρ(t) is drastically different when a trap is placed on a random node compared to the case of the trap being on the node with the maximum connectivity. For the latter case we find ρ(t) ∝ exp −At/N γ−2 γ−1 � kfor all γ> 2, where γ is the exponent of the degree distribution P (k) ∝ k �γ .

[1]  Albert-László Barabási,et al.  Evolution of Networks: From Biological Nets to the Internet and WWW , 2004 .

[2]  J. Douglas.  Aspects and applications of the random walk , 1995 .

[3]  Christos Gkantsidis,et al.  Random walks in peer-to-peer networks , 2004, IEEE INFOCOM 2004.

[4]  Cohen,et al.  Resilience of the internet to random breakdowns , 2000, Physical review letters.

[5]  P. Gács,et al.  Algorithms , 1992 .

[6]  George H. Weiss.  Contemporary Problems in Statistical Physics , 1987 .

[7]  G. Weiss.  Aspects and Applications of the Random Walk , 1994 .

[8]  Alessandro Vespignani,et al.  Evolution and Structure of the Internet: A Statistical Physics Approach , 2004 .

[9]  J. E. Kiefer,et al.  Trapping of random walks in two and three dimensions , 1984 .

[10]  Clifford Stein,et al.  Introduction to Algorithms, 2nd edition. , 2001 .

[11]  S. Havlin,et al.  Diffusion and Reactions in Fractals and Disordered Systems , 2000 .

[12]  Shlomo Havlin,et al.  Anomalous transport in scale-free networks. , 2005, Physical review letters.

[13]  Nitin H. Vaidya,et al.  Proceedings of the eleventh ACM international symposium on Mobile ad hoc networking and computing , 2010, MobiCom 2010.

[14]  W. Coffey,et al.  Diffusion and Reactions in Fractals and Disordered Systems , 2002 .

[15]  J. Klafter,et al.  Anomalous Size Dependence of Relaxational Processes , 1997, cond-mat/9904195.

[16]  R. Rosenfeld.  Nature , 2009, Otolaryngology--head and neck surgery : official journal of American Academy of Otolaryngology-Head and Neck Surgery.

[17]  Chen Avin,et al.  Efficient and robust query processing in dynamic environments using random walk techniques , 2004, Third International Symposium on Information Processing in Sensor Networks, 2004. IPSN 2004.

[18]  Béla Bollobás,et al.  Random Graphs , 1985 .

[19]  S. Havlin,et al.  Breakdown of the internet under intentional attack. , 2000, Physical review letters.

[20]  J. Stoyanov.  A Guide to First‐passage Processes , 2003 .

[21]  Reuven Cohen,et al.  Tomography of scale-free networks and shortest path trees. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Reuven Cohen,et al.  Numerical evaluation of the upper critical dimension of percolation in scale-free networks. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.