Public transport networks of fourteen cities of so far unexplored network size are analyzed in standardized graph representations: the simple graph of the network map, the bipartite graph of routes and stations, and both one mode projections of the latter. Special attention is paid to the inter-relations and spatial embedding of transport routes. This systematic approach reveals rich behavior beyond that of the ubiquitous scale-free complex network. We find strong evidence for structures in PTNs that are counter-intuitive and need to be explained, among these a pronounced diversity in the expression of typical network characteristics within the present sample of cities, a surprising geometrical behavior with respect to the two-dimensional geographical embedding and an unexpected attraction between transport routes. A simple model based on these observations reproduces many of the identified PTN properties by growing networks of attractive self-avoiding walks.

Public transport networks: empirical analysis and modeling

We use complex network concepts to analyze statistical properties of urban public transport networks (PTN). To this end, we present a comprehensive survey of the statistical properties of PTNs based on the data of fourteen cities of so far unexplored network size. Especially helpful in our analysis are different network representations. Within a comprehensive approach we calculate PTN characteristics in all of these representations and perform a comparative analysis. The standard network characteristics obtained in this way often correspond to features that are of practical importance to a passenger using public traffic in a given city. Specific features are addressed that are unique to PTNs and networks with similar transport functions (such as networks of neurons, cables, pipes, vessels embedded in 2D or 3D space). Based on the empirical survey, we propose a model that albeit being simple enough is capable of reproducing many of the identified PTN properties. A central ingredient of this model is a growth dynamics in terms of routes represented by self-avoiding walks.

We present a replica calculation and obtain analytically the size of a polymer chain with excluded volume interactions in quenched random media. We show that the size of the polymer is shrunk as the density of the impurities constituting the random medium is increased. We demonstrate that this is a general phenomenon. There are three regimes depending on the strength of the impurity density v. In the weak impurity density regime the polymer obeys the self‐avoiding statistics with its gyration radius R depending on its typical length L according to R∼L(D+2)/(d+2), where D is the dimension of the polymer and d is the space dimension. In the intermediate regime the polymer is in the unperturbed state where R∼L(2−D)/2. In the third regime of localization occurring for sufficiently large v, R∼v−(2−D)/[4D−(2−D)d] in the absence of three‐body interactions and R∼v−1/d LD/d in the presence of strong three‐body effects. We examine the effect of long‐range interactions of the type w‖r‖−α between the segments of the polymer separated by r on the localization and find that the polymer collapse is suppressed for realistic impurity densities [v<wR(d−α)/2].

Localization of a polymeric manifold in quenched random media

The authors study self-avoiding walks (SAW) on randomly diluted (quenched) lattices with direct configurational averaging over the moments of the SAW distribution function. A scaling function representation of RN, the average end-to-end distance of N-step walks, is studied here both for SAW on (a) the infinite percolation cluster and (b) any cluster. They have shown that, at the percolation threshold nu P= nu P(1- beta P/2 nu P), where beta P and nu P are the percolation order parameter and correlation length exponents respectively. The authors also propose a scaling function representation for the total number of N-step SAW configuration GN( approximately mu NNgamma -1) for infinite cluster averaging, which gives gamma P= gamma +d( nu P- nu ). For all cluster averaging gamma will remain unchanged.

Scaling theory for the statistics of self-avoiding walks on random lattices

The critical thermodynamics of the two-dimensional $N$-vector cubic and $MN$ models is studied within the field-theoretical renormalization group (RG) approach. The $\ensuremath{\beta}$ functions and critical exponents are calculated in the five-loop approximation and the RG series obtained are resummed using the Borel-Leroy transformation combined with the generalized Pad\'e approximant and conformal mapping techniques. For the cubic model, the RG flows for various $N$ are investigated. For $N=2$ it is found that the continuous line of fixed points running from the $XY$ fixed point to the Ising one is well reproduced by the resummed RG series and an account for the five-loop terms makes the lines of zeros of both $\ensuremath{\beta}$ functions closer to each another. For the cubic model with $N\ensuremath{\geqslant}3$, the five-loop contributions are shown to shift the cubic fixed point, given by the four-loop approximation, towards the Ising fixed point. This confirms the idea that the existence of the cubic fixed point in two dimensions under $Ng2$ is an artifact of the perturbative analysis. For the quenched dilute $O(M)$ models ($MN$ models with $N=0$) the results are compatible with a stable pure fixed point for $M\ensuremath{\geqslant}1$. For the $MN$ model with $M,N\ensuremath{\geqslant}2$ all the nonperturbative results are reproduced. In addition a new stable fixed point is found for moderate values of $M$ and $N$.

Critical behavior of two-dimensional cubic and MN models in the five-loop renormalization group approximation

Using a field theoretic method based on the replica trick, it is proved that the three-parameter renormalisation group for an n-vector model with quenched randomness reduces to a two-parameter one in the limit n to 0 which corresponds to self-avoiding walks (SAWS). This is also shown by the explicit calculation of the renormalisation-group recursion relations to second order in epsilon . From this reduction the author finds that SAWS on the random lattice are in the same universality class as SAWS on the regular lattice. By analogy with the case of the n-vector model with cubic anisotropy in the limit n to 1, the fixed-point structure of the n-vector model with randomness is analysed in the SAW limit, so that a physical interpretation of the unphysical fixed point is given. Corrections of the values of critical exponents of the unphysical fixed point published previously are also given.

Yup Kim

Papers

Renormalisation-group study of self-avoiding walks on the random lattice