%0 Book Section %1 statphys23_0795 %A Hase, M.O. %A Mendes, J.F.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 euclidean exact growing network results space statphys23 topic-11 %T Growing Network On Euclidean Space %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=795 %X Graph properties, like degree distributions, clustering, shortest path lengths, et cetera, have been intensively examined in structure and dynamics of complex networks. Many real networks, however, have an additional property of being embedded in a metric arrangement, where the geographic distance between vertices affects graph properties. The present work proposes a model for growing network that takes into account the Euclidean distance between vertices: the probability that two vertices connect each other decreases with their distance. Starting from a central node at time $t=0$, the network is made to grow toward $m$ directions -- or branches. At each time step, taken to be $1$, $m$ new vertices are added to the network, one at each branch. The positions of the vertices are such that they are located at distance $1$ from the previous vertex in the same branch, and also at distance $1$ from the vertices that were born at the same time in adjacent branches -- see Figure 1. Each vertex, born at time $t+1$, links to just one of the $mt+1$ previous vertices with probability $r^-\alpha/N$, where $r$ is the Euclidean distance between the vertices, and $N$ is the normalization, which depends on the size of the network ($t$), $\alpha$, and $m$. The parameter $\alpha$ tunes the strength of the range of interaction, and different behaviour of some graph properties are displayed as $\alpha$ varies on the non - negative real axis. The function $k(s,t)$, which is the mean degree of a vertex $s(<t)$ at time $t$ (the symmetry of the model allows one to concentrate in just one -- among $m$ -- branch) is extensively analyzed. To state some results, the mean degree, for $m=1$ and $1st$, behaves as $(1-\alpha)łn(t/s)$, $łn(t/s)$ and $2$ for $0łeq\alpha<1$, $\alpha=1$ and $\alpha>1$, respectively. For $1mst$, the result for the $m=1$ case is recovered, while for the other limit, $1stm$, the mean degree increases as $łn(t/s)$ (apart from a multiplicative term that depends on $\alpha$) for $0łeq\alpha2$. In the case $\alpha>2$, a special attention should be devoted to the parameter $mt^1-\alpha$. If $mt^1-\alpha1$, one still has a logarithmic behaviour of the mean degree, while for $mt^1-\alpha1$, $k(s,t)$ is limited and tends to $2$, as expected, as $\alpha$ increases. These later results stress the fact that the number of branches does affect the graph property of the network.

@incollection{statphys23_0795, abstract = {Graph properties, like degree distributions, clustering, shortest path lengths, et cetera, have been intensively examined in structure and dynamics of complex networks. Many real networks, however, have an additional property of being embedded in a metric arrangement, where the geographic distance between vertices affects graph properties. The present work proposes a model for growing network that takes into account the Euclidean distance between vertices: the probability that two vertices connect each other decreases with their distance. Starting from a central node at time $t=0$, the network is made to grow toward $m$ directions -- or branches. At each time step, taken to be $1$, $m$ new vertices are added to the network, one at each branch. The positions of the vertices are such that they are located at distance $1$ from the previous vertex in the same branch, and also at distance $1$ from the vertices that were born at the same time in adjacent branches -- see Figure 1. Each vertex, born at time $t+1$, links to just one of the $mt+1$ previous vertices with probability $r^{-\alpha}/N$, where $r$ is the Euclidean distance between the vertices, and $N$ is the normalization, which depends on the size of the network ($t$), $\alpha$, and $m$. The parameter $\alpha$ tunes the strength of the range of interaction, and different behaviour of some graph properties are displayed as $\alpha$ varies on the non - negative real axis. The function $\overline{k}(s,t)$, which is the mean degree of a vertex $s(<t)$ at time $t$ (the symmetry of the model allows one to concentrate in just one -- among $m$ -- branch) is extensively analyzed. To state some results, the mean degree, for $m=1$ and $1\ll s\ll t$, behaves as $(1-\alpha)\ln(t/s)$, $\ln(\ln t/\ln s)$ and $2$ for $0\leq\alpha<1$, $\alpha=1$ and $\alpha>1$, respectively. For $1\sim m\ll s\ll t$, the result for the $m=1$ case is recovered, while for the other limit, $1\ll s\ll t\ll m$, the mean degree increases as $\ln(t/s)$ (apart from a multiplicative term that depends on $\alpha$) for $0\leq\alpha\leq 2$. In the case $\alpha>2$, a special attention should be devoted to the parameter $mt^{1-\alpha}$. If $mt^{1-\alpha}\gg 1$, one still has a logarithmic behaviour of the mean degree, while for $mt^{1-\alpha}\ll 1$, $\overline{k}(s,t)$ is limited and tends to $2$, as expected, as $\alpha$ increases. These later results stress the fact that the number of branches does affect the graph property of the network.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Hase, M.O. and Mendes, J.F.F.}, biburl = {https://www.bibsonomy.org/bibtex/2e5a312012f005049b50e69f9bfe0445d/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {928b8da8bd07491f506a9705bd34782a}, intrahash = {e5a312012f005049b50e69f9bfe0445d}, keywords = {euclidean exact growing network results space statphys23 topic-11}, month = {9-13 July}, timestamp = {2007-06-20T10:16:29.000+0200}, title = {Growing Network On Euclidean Space}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=795}, year = 2007 }