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C. Borgs

Publications and source records attributed to C. Borgs.

12 recordsLinked to original sources
arXiv2011

Finding undetected protein associations in cell signaling by belief propagation

External information propagates in the cell mainly through signaling cascades and transcriptional activation, allowing it to react to a wide spectrum of environmental changes. High throughput experiments identify numerous molecular components of such cascades that may, however, interact through unknown partners. Some of them may be detected using data coming from the integration of a protein-protein interaction network and mRNA expression profiles. This inference problem can be mapped onto the problem of finding appropriate optimal connected subgraphs of a network defined by these datasets. The optimization procedure turns out to be computationally intractable in general. Here we present a new distributed algorithm for this task, inspired from statistical physics, and apply this scheme to alpha factor and drug perturbations data in yeast. We identify the role of the COS8 protein, a member of a gene family of previously unknown function, and validate the results by genetic experiments. The algorithm we present is specially suited for very large datasets, can run in parallel, and can be adapted to other problems in systems biology. On renowned benchmarks it outperforms other algorithms in the field.

q-bio.MN↗
arXiv2009

Limits of randomly grown graph sequences

Motivated in part by various sequences of graphs growing under random rules (like internet models), convergent sequences of dense graphs and their limits were introduced by Borgs, Chayes, Lovász, Sós and Vesztergombi and by Lovász and Szegedy. In this paper we use this framework to study one of the motivating class of examples, namely randomly growing graphs. We prove the (almost sure) convergence of several such randomly growing graph sequences, and determine their limit. The analysis is not always straightforward: in some cases the cut distance from a limit object can be directly estimated, in other case densities of subgraphs can be shown to converge.

math.CO↗
arXiv2008

Statistical Mechanics of Steiner trees

The Minimum Weight Steiner Tree (MST) is an important combinatorial optimization problem over networks that has applications in a wide range of fields. Here we discuss a general technique to translate the imposed global connectivity constrain into many local ones that can be analyzed with cavity equation techniques. This approach leads to a new optimization algorithm for MST and allows to analyze the statistical mechanics properties of MST on random graphs of various types.

cond-mat.stat-mech↗
arXiv2007

Fitting the WHOIS Internet data

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.

cs.NI↗
arXiv2007

Convergent Sequences of Dense Graphs I: Subgraph Frequencies, Metric Properties and Testing

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.

math.CO↗
arXiv2005

Degree Distribution of Competition-Induced Preferential Attachment 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.

cond-mat.dis-nn↗
arXiv2004

Competition-Induced Preferential Attachment

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.

cond-mat.stat-mech↗
arXiv2003

Phase Diagram for the Constrained Integer Partitioning Problem

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.

cond-mat.dis-nn↗
arXiv1995

SURFACE INDUCED FINITE-SIZE EFFECTS FOR FIRST ORDER PHASE TRANSITIONS

We consider classical lattice models describing first-order phase transitions, and study the finite-size scaling of the magnetization and susceptibility. In order to model the effects of an actual surface in systems like small magnetic clusters, we consider models with free boundary conditions. For a field driven transition with two coexisting phases at the infinite volume transition point $h=h_t$, we prove that the low temperature finite volume magnetization $m_{\free}(L,h)$ per site in a cubic volume of size $L^d$ behaves like $m_\free(L,h)=\frac{m_++m_-}2 + \frac{m_+-m_-}2 \tanh \bigl(\frac{m_+-m_-}2\,L^d\, (h-h_χ(L))\bigr)+O(1/L)$, where $h_χ(L)$ is the position of the maximum of the (finite volume) susceptibility and $m_\pm$ are the infinite volume magnetizations at $h=h_t+0$ and $h=h_t-0$, respectively. We show that $h_χ(L)$ is shifted by an amount proportional to $1/L$ with respect to the infinite volume transitions point $h_t$ provided the surface free energies of the two phases at the transition point are different. This should be compared with the shift for periodic boun\- dary conditons, which for an asymmetric transition with two coexisting phases is proportional only to $1/L^{2d}$. One also consider the position $h_U(L)$ of the maximum of the so called Binder cummulant $U_\free(L,h)$. While it is again shifted by an amount proportional to $1/L$ with respect to the infinite volume transition point $h_t$, its shift with respect to $h_χ(L)$ is of the much smaller order $1/L^{2d}$. We give explicit formulas for the proportionality factors, and show that, in the leading $1/L^{2d}$ term, the relative shift is the same as that for periodic boundary conditions.

cond-mat↗
arXiv1995

THE STAGGERED CHARGE-ORDER PHASE OF THE EXTENDED HUBBARD MODEL

We study the phase diagram of the extended Hubbard model in the atomic limit. At zero temperature, the phase diagram decomposes into six regions: three with homogeneous phases (characterized by particle densities $ρ=0$, 1, and 2 and staggered charge density $Δ=0$) and three with staggered phases (characterized by the densities $ρ=\frac12$, 1, and $\frac32$ and staggered densities $|Δ|=\frac12$, 1, and $\frac12$). Here we use Pirogov-Sinai theory to analyze the details of the phase diagram of this model at low temperatures. In particular, we show that for any sufficiently low nonzero temperature the three staggered regions merge into one staggered region $S$, without any phase transitions (analytic free energy and staggered order parameter $Δ$) within $S$.

cond-mat↗
arXiv1994

The covariance matrix of the potts model: A random cluster analysis

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.

adap-org↗