arXiv subjects
Publications and source records attributed to J. T. Chayes.
We consider the RIPE WHOIS Internet data as characterized by the Cooperative Association for Internet Data Analysis (CAIDA), and show that the Tempered Preferential Attachment model [1] provides an excellent fit to this data. [1] D'Souza, Borgs, Chayes, Berger and Kleinberg, to appear PNAS USA, 2007.
We consider sequences of graphs and define various notions of convergence related to these sequences: ``left convergence'' defined in terms of the densities of homomorphisms from small graphs into the graphs of the sequence, and ``right convergence'' defined in terms of the densities of homomorphisms from the graphs of the sequence into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs, and for graphs with nodeweights and edgeweights. One of the main steps here is the introduction of a cut-distance comparing graphs, not necessarily of the same size. We also show how these notions of convergence provide natural formulations of Szemeredi partitions, sampling and testing of large graphs.
We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs between two competing metrics, and show that this family is equivalent to a family of preferential attachment random graph models with upper cutoffs. This is the first explanation of how preferential attachment can arise from a more basic underlying mechanism of local competition. We rigorously determine the degree distribution for the family of random graph models, showing that it obeys a power law up to a finite threshold and decays exponentially above this threshold. We also rigorously analyze a generalized version of our graph process, with two natural parameters, one corresponding to the cutoff and the other a ``fertility'' parameter. We prove that the general model has a power-law degree distribution up to a cutoff, and establish monotonicity of the power as a function of the two parameters. Limiting cases of the general model include the standard preferential attachment model without cutoff and the uniform attachment model.
Models based on preferential attachment have had much success in reproducing the power law degree distributions which seem ubiquitous in both natural and engineered systems. Here, rather than assuming preferential attachment, we give an explanation of how it can arise from a more basic underlying mechanism of competition between opposing forces. We introduce a family of one-dimensional geometric growth models, constructed iteratively by locally optimizing the tradeoffs between two competing metrics. This family admits an equivalent description as a graph process with no reference to the underlying geometry. Moreover, the resulting graph process is shown to be preferential attachment with an upper cutoff. We rigorously determine the degree distribution for the family of random graph models, showing that it obeys a power law up to a finite threshold and decays exponentially above this threshold. We also introduce and rigorously analyze a generalized version of our graph process, with two natural parameters, one corresponding to the cutoff and the other a ``fertility'' parameter. Limiting cases of this process include the standard Barabasi-Albert preferential attachment model and the uniform attachment model. In the general case, we prove that the process has a power law degree distribution up to a cutoff, and establish monotonicity of the power as a function of the two parameters.
We consider the problem of partitioning $n$ integers into two subsets of given cardinalities such that the discrepancy, the absolute value of the difference of their sums, is minimized. The integers are i.i.d. random variables chosen uniformly from the set $\{1,...,M\}$. We study how the typical behavior of the optimal partition depends on $n,M$ and the bias $s$, the difference between the cardinalities of the two subsets in the partition. In particular, we rigorously establish this typical behavior as a function of the two parameters $κ:=n^{-1}\log_2M$ and $b:=|s|/n$ by proving the existence of three distinct ``phases'' in the $κb$-plane, characterized by the value of the discrepancy and the number of optimal solutions: a ``perfect phase'' with exponentially many optimal solutions with discrepancy 0 or 1; a ``hard phase'' with minimal discrepancy of order $Me^{-Θ(n)}$; and a ``sorted phase'' with an unique optimal partition of order $Mn$, obtained by putting the $(s+n)/2$ smallest integers in one subset. Our phase diagram covers all but a relatively small region in the $κb$-plane. We also show that the three phases can be alternatively characterized by the number of basis solutions of the associated linear programming problem, and by the fraction of these basis solutions whose $\pm 1$-valued components form optimal integer partitions of the subproblem with the corresponding weights. We show in particular that this fraction is one in the sorted phase, and exponentially small in both the perfect and hard phases, and strictly exponentially smaller in the hard phase than in the perfect phase. Open problems are discussed, and numerical experiments are presented.
We consider the covariance matrix $G^{mn}(x-y)$ of the d-dimensional q-states Potts model, rewriting it in terms of the connectivity, the finite-cluster connectivity and the infinite-cluster covariance in the random cluster repre- sentation of Fortuin and Kasteleyn. In any of the $q$ ordered phases, we show that the matrix $G^{mn}(x-y)$ has one tivial eigenvalue 0, one simple eigen- value $G_{\wir}^{(1)}(x-y)$ and one ($q-2$)-fold degenerate eigenvalue $G_{\wir}^{(2)}(x-y)$. Furthermore, we identify the eigenvalues both in terms of representations of the unbroken symmetry group of the model, and in terms of connectivities and cluster covariances, thereby attributing algebraic signifi- cance to these stochastic geometric quantities. In addition to establishing the existence of the correlation lengths $ξ_{\wir}^{(1)}$ and $ξ_{\wir}^{(2)}$ corresponding to $G_{\wir}^{(1)}(x-y)$ and $G_{\wir}^{(2)}(x-y)$, we show that $ξ_{\wir}^{(1)}(β)\geq ξ_{\wir}^{(2)}(β)$ for all inverse tempera- tures $β$. For dimension $d=2$ and $q \geq 1$, we establish a duality relation between $ξ_{\wir}^{(2)}$ and $ξ_{\free}$, the correlation length of the two-point function with free boundary conditions: We show $ξ_{\wir}^{(2)}(β) = \frac{1}{2} ξ_{\free}(β^\ast)$ for all $β\geq β_o$, where $β^\ast$ is the dual inverse temperature and $β_o$ is the self-dual point. In order to prove the above results, we introduce two new inequalities. The first is similar to the FKG inequality, but holds for events which are neither increasing nor decreasing, and replaces independence in the standard percolation model; the second replaces the van den Berg - Kesten inequality.