Random field Ising model and community structure in complex networks

We propose a method to determine the community structure of a complex network. In this method the ground state problem of a ferromagnetic random field Ising model is considered on the network with the magnetic field B_s = +∞, B_t = -∞, and B_i≠s,t=0 for a node pair s and t. The ground state problem is equivalent to the so-called maximum flow problem, which can be solved exactly numerically with the help of a combinatorial optimization algorithm. The community structure is then identified from the ground state Ising spin domains for all pairs of s and t. Our method provides a criterion for the existence of the community structure, and is applicable equally well to unweighted and weighted networks. We demonstrate the performance of the method by applying it to the Barabási-Albert network, Zachary karate club network, the scientific collaboration network, and the stock price correlation network. (Ising, Potts, etc.)     

Random pseudofractal scale-free networks with small-world effect

A random pseudofractal network (RPN) is generated by a recursive growing rule. The RPN is of the scale-free feature and small-world effect. We obtain the theoretical results of power-law exponent γ=3, clustering coefficient C=3π²-19≈ 0.74, and a proof that the mean distance increases no faster than ln N, where N is the network size. These results agree with the numerical simulation very well. In particular, we explain the property of growth and preferential attachment in RPNs. And the properties of a class of general RPNs are discussed in the end.     

Definitions of equilibrium in network formation games

We examine a variety of stability and equilibrium definitions that have been used to study the formation of social networks among a group of players. In particular we compare variations on three types of definitions: those based on a pairwise stability notion, those based on the Nash equilibria of a link formation game, and those based on equilibria of a link formation game where transfers are possible.Bloch is also affiliated with the University of Warwick.     

Exploitation of the Fiber Capacity in Optical Networks Using Time, Frequency, and Space Division Multiple Accesses

In this work we investigate how to obtain very high capacity transmissions in optical networks taking into account the limitations due to the physical channel. We consider both the case in which all the users are connected by a star coupler and the case in which the users are directly connected by the network topology. As a reference, we consider a ring network and a Shuffle Multihop Network (SMN). The use of optical systems to implement high-capacity networks is numerically investi gated by means of numerical simulations taking into consideration the channel limitations due to the chromatic dispersion, the Kerr effect, and the amplified spontaneous emission (ASE) noise of the optical amplifiers. In our model, we consider that the signal, during the routing process that is performed at the user position, undergoes only an attenuation. We suppose the use of intensity modulated signals and receivers with direct detection. Packet switching and digital transmission are assumed with soliton and conventional nonreturn to zero signals. Both wave length and time division multiple accesses are considered. The results show that, in the case of the Time Division Multiple Access (TDMA) technique, the use of a star coupler to connect the users reduces the capacity of a network with respect to the case in which a direct connection of the users is used. This is due to the strong power fluctuations that are present during the signal propagation and to the large quantity of accumulated ASE noise. On the other hand, the use of a star coupler shows the advantage to being easily reconfigurable. The Wavelength Divison Multiple Access (WDMA) technique permits us to achieve higher capacities with respect to the TDMA. This is due to the fact that in the propagation conditions, due to the presence of a star coupler, high bit rate signals are strongly degraded. On the other hand, several low bit rate signals operating at different wavelengths can propagate with a low power level, avoiding strong degradation due to the Four Wave Mixing (FWM) effect. Among the topologies considered in this work, the SMN is the one that generally permits us to reach the highest throughput because in the SMN the signal hops in a limited number of Network Interface Units (NIUs) before reaching the final destination.     

An Analysis of the Transition Zone Between the Various Scaling Regimes in the Small-World Model

We analyse the so-called small-world network model (originally devised by Strogatz and Watts), treating it, among other things, as a case study of non-linear coupled difference or differential equations. We derive a system of evolution equations containing more of the previously neglected (possibly relevant) non-linear terms. As an exact solution of this entangled system of equations is out of question we develop a (as we think, promising) method of enclosing the “exact” solutions for the expected quantities by upper and lower bounds, which represent solutions of a slightly simpler system of differential equation. Furthermore we discuss the relation between difference and differential equations and scrutinize the limits of the spreading idea for random graphs. We then show that there exists in fact a “broad” (with respect to scaling exponents) crossover zone, smoothly interpolating between linear and logarithmic scaling of the diameter or average distance. We are able to corroborate earlier findings in certain regions of phase or parameter space (as e.g. the finite size scaling ansatz) but find also deviations for other choices of the parameters. Our analysis is supplemented by a variety of numerical calculations, which, among other things, quantify the effect of various approximations being made. With the help of our analytical results we manage to calculate another important network characteristic, the (fractal) dimension, and provide numerical values for the case of the small-world network.     

ON THE BOTTLENECK CAPACITY EXPANSION PROBLEMS ON NETWORKS

This article considers a class of bottleneck capacity expansion problems. Such problems aim to enhance bottleneck capacity to a certain level with minimum cost. Given a network G(V, A, C) consisting of a set of nodes V= {v1,v2,…,vn},a set of arcs A ■ {(vi, vj) | i=1,2,…, n;j=1,2,…,n} and a capacity vector C. The component cij of C is the capacity of arc (vi ,vj). Define the capacity of a subset A' of A as the minimum capacity of the arcs in A, the capacity of a family F of subsets of A is the maximum capacity of its members. There are two types of expanding models. In the arc-expanding model, the unit cost to increase the capacity of arc (vi, vj) is wij. In the node-expanding model, it is assumed that the capacities of all arcs (vi,vj) which start at the same node vi should be increased by the same amount and that the unit cost to make such expansion is wi. This article considers three kinds of bottleneck capacity expansion problems (path, spanning arborescence and maximum flow) in both expanding models. For each kind of expansion problems, this article discusses the characteristics of the problems and presents several results on the complexity of the problems.     

Glorious uncertainty—challenges for network design

This article discusses the challenges that face the crystal engineer in the deliberate design of new network structures. These include control over ligand and metal coordination geometry, selection of network topology from a number of possibilities which all have the same connectivity, and control of methods of increasing packing efficiency, including interpenetration (both number of nets (including only one) and topology of interpenetration). These variables can lead to polymorphism and related phenomena, the bane of crystal engineers. Templation by counterions, guest molecules and/or solvents can also lead to unpredictable results.     

关于3连通图的容错直径和宽直径

容错直径和宽直径是度量网络可靠性和有效性的重要参数．对任意k连通图，它的容错直径Dk不超过宽直径dk,本证明：当D2=2时，d3≤max{D， l，2D3-2}；当D2≥3时，d3≤(D2-1)[2(D2-1)(D3-1)-D2-2] 1．     

Minority opinion spreading in random geometry

The dynamics of spreading of the minority opinion in public debates (a reform proposal, a behavior change, a military retaliation) is studied using a diffusion reaction model. People move by discrete step on a landscape of random geometry shaped by social life (offices, houses, bars, and restaurants). A perfect world is considered with no advantage to the minority. A one person-one argument principle is applied to determine locally individual mind changes. In case of equality, a collective doubt is evoked which in turn favors the Status Quo. Starting from a large in favor of the proposal initial majority, repeated random size local discussions are found to drive the majority reversal along the minority hostile view. Total opinion refusal is completed within few days. Recent national collective issues are revisited. The model may apply to rumor and fear propagation. Received 23 January 2002     

Bounds for aggregating nodes in network problems

It is often necessary or desirable in practice to replace a large, detailed optimization model with a smaller, approximate model. It would be useful to have bounds on the error resulting from this process of aggregation. This paper develops such bounds for a class of linear minimum-cost network-flow problems, where groups of nodes in a large problem are replaced by aggregate nodes. Two types of bounds are derived: A priori bounds are available before solving the aggregated problem; a posteriori bounds, which are generally tighter, may be computed afterwards.