%0 Book Section %1 statphys23_1096 %A Zara, R.A. %A Michel, N.F. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K optimum overloads paths percolation statphys23 topic-11 %T Overload dynamics in scale free network %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1096 %X Many real complex networks show heterogeneous structures with power-law degree distribution $P(k) k^-\gamma$, where $k$ is the number of links of a randomly chosen node in the network and $\gamma$ is the scaling exponent. This algebraic distribution means that, in contrast to random networks, the probability for a node to possess a large number of links is not exponentially small. Due the ubiquity of scale-free networks in natural and manmade systems, problems related to the security of these networks and their resilience have attracted a great interest. A myriad of important aspects concernig complex networks, including disturbances in power transmission systems, effects of network growth, cascading failures triggered by intentional attacks, avalanche size distributions and congestion instabilities have been discussed in the literature. In some previous works the load was introduced to address the pattern of transport by considering both links and nodes as identical in terms of their functional roles in the network. Such studies can be generalized investigating the heterogeneity of elements by introducing weights (e.g. cost) to the links. Congestion effects may be taken into account by investigating the weight of a link as a function of its cumulative load (called cost function), which is common in transport networks. Using simple models of load distribution the congestion effects may be described as function of time. In this scope we investigate the disruption phenomenon in transport network, i.e., how the nodes become unavailable after removing some overloaded links. This work differs on the previous in the sense that the overload of the links is caused by a dynamical process of overload. The transport properties as the size of giant connected cluster, the length of the minimum paths and the optimal paths between a pair of nodes (which are used to define the efficience of the network) were evaluated as a function of time. We start with a standard Barabasi-Albert scale-free network of $N$ nodes and mean conectivity $łeftk \right= 4$. At $t=0$ an initial (uniform distributed in the interval $ łeft0,1 \right)$ load and a maximum load capacity $C$ is assigned to each link of the network . In each time step an additional load $N p $ (where $p$ is the probability of a node to be chosen to receive the load) is introduced in the system. The load $\beta$ assigned to a node is distributed among its $k$ links in a such way that each link receives a fraction of load inversely proportional to its own load. If the load atributed to a link exceeds its load capacity, the link becomes overloaded. In this case the link becomes unavailable and its contribution to the network transport properties is null. Due the eventual overload of links nodes or clusters of nodes may be isolated and the network may be broken in many isolated clusters. One can define different load introduction strategies by choosing different functions to the probability $p$. Here we follow two strategies: (a) an uniform random distribution of loads and (b) a Cohen-like immunization strategy (one node is selected with a uniform probability $p$ and one of its first neighbours, randomly selected, receives the load). Both strategies may be classified as local strategies but the resulting effects are qualitatively different. The figure shows a sample of obtained results. The quantities plotted are the size of the giant connected cluster $S$, the length of minimum paths connecting a pair of nodes measured as the minimum number of links connecting them ($ł_min$) and the optimal paths ($b_min$) connecting a pair of nodes evaluated as the path in which the cumulative load is minimum. All these quantities are normalized by their respective values at $t=0$. Evaluating the physical quantities as a function of time we observe that for the random strategy (figure a) there is a crossover from a fully connected cluster to a non-connected state in the sense that all links become unavailable. On the other hand, following the Cohen-like strategy we found an abrupt change in transport properties which may be interpreted as a percolation-like transition induced by the cumulative process of load along the time.

@incollection{statphys23_1096, abstract = {Many real complex networks show heterogeneous structures with power-law degree distribution $P(k) \propto k^{-\gamma}$, where $k$ is the number of links of a randomly chosen node in the network and $\gamma$ is the scaling exponent. This algebraic distribution means that, in contrast to random networks, the probability for a node to possess a large number of links is not exponentially small. Due the ubiquity of scale-free networks in natural and manmade systems, problems related to the security of these networks and their resilience have attracted a great interest. A myriad of important aspects concernig complex networks, including disturbances in power transmission systems, effects of network growth, cascading failures triggered by intentional attacks, avalanche size distributions and congestion instabilities have been discussed in the literature. In some previous works the load was introduced to address the pattern of transport by considering both links and nodes as identical in terms of their functional roles in the network. Such studies can be generalized investigating the heterogeneity of elements by introducing weights (e.g. cost) to the links. Congestion effects may be taken into account by investigating the weight of a link as a function of its cumulative load (called cost function), which is common in transport networks. Using simple models of load distribution the congestion effects may be described as function of time. In this scope we investigate the disruption phenomenon in transport network, i.e., how the nodes become unavailable after removing some overloaded links. This work differs on the previous in the sense that the overload of the links is caused by a dynamical process of overload. The transport properties as the size of giant connected cluster, the length of the minimum paths and the optimal paths between a pair of nodes (which are used to define the efficience of the network) were evaluated as a function of time. We start with a standard Barabasi-Albert scale-free network of $N$ nodes and mean conectivity $\left\langle k \right\rangle = 4$. At $t=0$ an initial (uniform distributed in the interval $ \left[0,1 \right])$ load and a maximum load capacity $C$ is assigned to each link of the network . In each time step an additional load $\beta N p $ (where $p$ is the probability of a node to be chosen to receive the load) is introduced in the system. The load $\beta$ assigned to a node is distributed among its $k$ links in a such way that each link receives a fraction of load inversely proportional to its own load. If the load atributed to a link exceeds its load capacity, the link becomes overloaded. In this case the link becomes unavailable and its contribution to the network transport properties is null. Due the eventual overload of links nodes or clusters of nodes may be isolated and the network may be broken in many isolated clusters. One can define different load introduction strategies by choosing different functions to the probability $p$. Here we follow two strategies: (a) an uniform random distribution of loads and (b) a Cohen-like immunization strategy (one node is selected with a uniform probability $p$ and one of its first neighbours, randomly selected, receives the load). Both strategies may be classified as local strategies but the resulting effects are qualitatively different. The figure shows a sample of obtained results. The quantities plotted are the size of the giant connected cluster $S$, the length of minimum paths connecting a pair of nodes measured as the minimum number of links connecting them ($\l_{min}$) and the optimal paths ($b_{min}$) connecting a pair of nodes evaluated as the path in which the cumulative load is minimum. All these quantities are normalized by their respective values at $t=0$. Evaluating the physical quantities as a function of time we observe that for the random strategy (figure a) there is a crossover from a fully connected cluster to a non-connected state in the sense that all links become unavailable. On the other hand, following the Cohen-like strategy we found an abrupt change in transport properties which may be interpreted as a percolation-like transition induced by the cumulative process of load along the time.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Zara, R.A. and Michel, N.F.}, biburl = {https://www.bibsonomy.org/bibtex/20ff17eec86fdaab7e15f9ada8b15eb16/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {0beb950c4e25c66da44d0adb27744691}, intrahash = {0ff17eec86fdaab7e15f9ada8b15eb16}, keywords = {optimum overloads paths percolation statphys23 topic-11}, month = {9-13 July}, timestamp = {2007-06-20T10:16:39.000+0200}, title = {Overload dynamics in scale free network}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1096}, year = 2007 }