Abstract:
The smallest positive eigenvalue of the Laplacian of a network is called the spectral gap and characterizes various dynamics on networks. We propose mathematical programming methods to maximize the spectral gap of a given network by removing a fixed number of nodes. We formulate relaxed versions of the original problem using semidefinite programming and apply them to example networks.

Abstract:
Preferential attachment is a popular model of growing networks. We consider a generalized model with random node removal, and a combination of preferential and random attachment. Using a high-degree expansion of the master equation, we identify a topological phase transition depending on the rate of node removal and the relative strength of preferential vs. random attachment, where the degree distribution goes from a power law to one with an exponential tail.

Abstract:
We study the resilience of complex networks against attacks in which nodes are targeted intelligently, but where disabling a node has a cost to the attacker which depends on its degree. Attackers have to meet these costs with limited resources, which constrains their actions. A network's integrity is quantified in terms of the efficacy of the process that it supports. We calculate how the optimal attack strategy and the most attack-resistant network degree statistics depend on the node removal cost function and the attack resources. The resilience of networks against intelligent attacks is found to depend strongly on the node removal cost function faced by the attacker. In particular, if node removal costs increase sufficiently fast with the node degree, power law networks are found to be more resilient than Poissonian ones, even against optimized intelligent attacks.

Abstract:
A phenomenological model that considers different secondary $id_{xy}$ gap amplitudes and quasiparticle effective charges for each nodal direction, is proposed to explain the observed anisotropic node removal by a magnetic field in high-$T_{c}$ cuprates. Two independent parity-breaking condensates $<\bar{\Psi_{i}}\bigwedge{\Psi}_{i}>$ develop, implying the induction of a magnetic moment per each nodal direction. The model outcomes are in agreement with the experimentally found relation $\Delta_{i}\sim\sqrt{B}$. The secondary gap vanishes through a second-order phase transition at a critical temperature whose value for underdoped nodal-oriented YBCO is estimated to be $\sim 6K$ for a field of 15 T.

Abstract:
The connectivity structure of a network can be very sensitive to removal of certain nodes in the network. In this paper, we study the sensitivity of the largest component size to node removals. We prove that minimizing the largest component size is equivalent to solving a matrix one-norm minimization problem whose column vectors are orthogonal and sparse and they form a basis of the null space of the associated graph Laplacian matrix. A greedy node removal algorithm is then proposed based on the matrix one-norm minimization. In comparison with other node centralities such as node degree and betweenness, experimental results on US power grid dataset validate the effectiveness of the proposed approach in terms of reduction of the largest component size with relatively few node removals.

Abstract:
The identification of modular structures is essential for characterizing real networks formed by a mesoscopic level of organization where clusters contain nodes with a high internal degree of connectivity. Many methods have been developed to unveil community structures, but only a few studies have probed their suitability in incomplete networks. Here we assess the accuracy of community detection techniques in incomplete networks generated in sampling processes. We show that the walktrap and fast greedy algorithms are highly accurate for detecting the modular structure of incomplete complex networks even if many of their nodes are removed. Furthermore, we implemented an approach that improved the time performance of the walktrap and fast greedy algorithms, while retaining the accuracy rate in identifying the community membership of nodes. Taken together our results show that this new approach can be applied to speed up virtually any community detection method in dense complex networks, as it is the case of similarity networks.

Abstract:
A network's transmission capacity is the maximal rate of traffic inflow that the network can handle without causing congestion. Here we study how to enhance this quantity by redistributing the capability of individual nodes while preserving the total sum of node capability. We propose a practical and effective node-capability allocation scheme which allocates a node's capability based on the local knowledge of the node's connectivity. We show the scheme enhances the transmission capacity by two orders of magnitude for networks with heterogenous structures.

Abstract:
A scaled Chebyshev node distribution is studied in this paper. It is proved that the node distribution is optimal for interpolation in $C_M^{s+1}[-1,1]$, the set of $(s+1)$-time differentiable functions whose $(s+1)$-th derivatives are bounded by a constant $M>0$. Node distributions for computing spectral differentiation matrices are proposed and studied. Numerical experiments show that the proposed node distributions yield results with higher accuracy than the most commonly used Chebyshev-Gauss-Lobatto node distribution.

Abstract:
We present a method to determine the nodal structure of the energy gap of unconventional superconductors such as high $T_c$ materials. We show how nonlinear electrodynamics phenomena in the Meissner regime, arising from the presence of lines on the Fermi surface where the superconducting energy gap is very small or zero, can be used to perform ``node spectroscopy'', that is, as a sensitive bulk probe to locate the angular position of those lines. In calculating the nonlinear supercurrent response, we include the effects of orthorhombic distortion and $a-b$ plane anisotropy. Analytic results presented demonstrate a systematic way to experimentally distinguish order parameters of different symmetries, including cases with mixed symmetry (for example, $d+s$ and $s+id$). We consider, as suggested by various experiments, order parameters with predominantly $d$-wave character, and describe how to determine the possible presence of other symmetries. The nonlinear magnetic moment displays a distinct behavior if nodes in the gap are absent but regions with small, finite, values of the energy gap exist.

Abstract:
We prove the following for a bounded convex planar domain that is symmetric with respect to both coordinate axes. Consider a centered rectangle with sides parallel to the axes that strictly contains the domain. If the domain is not a certain kind of rectangle, the spectral gap of the domain is larger than the spectral gap of the rectangle. We also provide explicit lower bounds for the differnce between the gaps.