American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

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Networks & Heterogeneous Media

September 2012 , Volume 7 , Issue 3

Special Issue
on Mesoscales and Evolution in Complex Networks: Applications and Related Topics
Guest Editors: Regino Criado, Rosa M. Benito, Miguel Romance and Juan C. Losada

Special Issue Papers: 363-481; Regular Papers: 483-582

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2012, 7(3): i-iii doi: 10.3934/nhm.2012.7.3i +[Abstract](2103) +[PDF](177.2KB)
Abstract:
The study of networks has become one of the paradigms of the science of complexity as well as a fascinating branch of research in applied mathematics, physics, engineering, sociology, biology and science in general. Different systems such as transport networks (underground, train, airline networks, road networks), communication networks (computer servers, Internet, online social networks), neural networks (neural interaction networks and brain networks), biochemical networks (metabolic, protein and genomic networks), trophic networks, social community networks, marketing and recommendation networks, other infrastructure networks (electric power grids, water supply networks) and many others (including the World Wide Web)([1],[3],[4],[7],[8],[9],[10]) are known to have behavioral and structural characteristics in common, and they can be studied by using non-linear mathematical techniques and computer modeling approaches. The interest on complex networks has certainly been promoted by the optimized rating of computing facilities, and by the availability of data on large real networks (including the World Wide Web, cortical networks, citation networks from Scientific Citation Index and online social networks). This focused section is characterized for emphasizing the latest applications of complex networks rather than the theoretical aspects, but covering several aspects as topological properties, algorithms and computation tools, models of interactions between complex systems, synchronization, control and some other related topics.

2012, 7(3): 363-371 doi: 10.3934/nhm.2012.7.363 +[Abstract](2421) +[PDF](5390.6KB)
Abstract:
Serendipity is defined as fortunate discoveries made by chance. In this work we explore the idea that topological measures of a person's social network could be an indicator about how likely that person is to experience fortunate discoveries.
2012, 7(3): 373-384 doi: 10.3934/nhm.2012.7.373 +[Abstract](2251) +[PDF](448.1KB)
Abstract:
The centrality and efficiency measures of an undirected network $G$ were shown by the authors to be strongly related to the respective measures on the associated line graph $L(G)$. In this note we extend this study to a directed network $\vec{G}$ and its associated directed network $\vec{L}(\vec{G})$. The Bonacich centralities of these two networks are shown to be related in a surprisingly simpler manner than in the non directed case. Efficiency is also considered and the corresponding relations established. In addition, an estimation of the clustering coefficient of $\vec{L}(\vec{G})$ is given in terms of the clustering coefficient of $\vec{G}$, and by means of an example we show that a reverse estimation cannot be expected.
Given a non directed graph $G$, there is a natural way to obtain from it a directed line graph, namely $\vec{L}(D(G))$, where the directed graph $D(G)$ is obtained from $G$ in the usual way. With this approach the authors estimate some parameters of $\vec{L}(D(G))$ in terms of the corresponding ones in $L(G)$. Particularly, we give an estimation of the norm difference between the centrality vectors of $\vec{L}(D(G))$ and $L(G)$ in terms of the Collatz-Sinogowitz index (which is a measure of the irregularity of $G$). Analogous estimations are given for the efficiency measures. The results obtained strongly suggest that for a given non directed network $G$, the directed line graph $\vec{L}(D(G))$ captures more adequately the properties of $G$ than the non directed line graph $L(G)$.
2012, 7(3): 385-397 doi: 10.3934/nhm.2012.7.385 +[Abstract](2903) +[PDF](4290.4KB)
Abstract:
Trade is a fundamental pillar of economy and a form of social organization. Its empirical characterization at the worldwide scale is represented by the World Trade Web (WTW), the network built upon the trade relationships between the different countries. Several scientific studies have focused on the structural characterization of this network, as well as its dynamical properties, since we have registry of the structure of the network at different times in history. In this paper we study an abstract scenario for the development of global crises on top of the structure of connections of the WTW. Assuming a cyclic dynamics of national economies and the interaction of different countries according to the import-export balances, we are able to investigate, using a simple model of pulse-coupled oscillators, the synchronization phenomenon of crises at the worldwide scale. We focus on the level of synchronization measured by an order parameter at two different scales, one for the global system and another one for the mesoscales defined through the topology. We use the WTW network structure to simulate a network of Integrate-and-Fire oscillators for six different snapshots between years 1950 and 2000. The results reinforce the idea that globalization accelerates the global synchronization process, and the analysis at a mesoscopic level shows that this synchronization is different before and after globalization periods: after globalization, the effect of communities is almost inexistent.
2012, 7(3): 399-413 doi: 10.3934/nhm.2012.7.399 +[Abstract](2251) +[PDF](3137.1KB)
Abstract:
Adam Smith is considered the father of modern economics. His research on the Wealth of Nations [10] is the first scientific work that theorized about the complexity of economic systems and how an invisible hand self-regulates markets and their behavior. In this way, we study international trade markets as complex networks. We analyze their topological properties, structure and temporal dynamics based on actual data. Our main premise states that trade networks are bipartite in nature because importers and exporters play a different role in the system. We apply a methodology developed for mutualistic ecosystems, finding minor gaps in it. We address such gaps by using well-known techniques from other related scientific work. The evidence supports the fact that our premise is a realistic hypothesis.
2012, 7(3): 415-428 doi: 10.3934/nhm.2012.7.415 +[Abstract](2473) +[PDF](2072.1KB)
Abstract:
The goal of this research is to estimate different parameters in the urban bus and the subway networks of Madrid. The obtained results will allow learning more about both types of networks: modularity, most important stops, sensitivity in the district networks (districts with highest and lowest sensitivity), bus line concentration by detected communities, communication capacity for these networks (districts with the greatest and less number of inner and external communications), and relation between network and dweller density by district. This study can help to improve the transport networks: reducing the district sensitivity, adding new stops or routes, etc.
2012, 7(3): 429-440 doi: 10.3934/nhm.2012.7.429 +[Abstract](2697) +[PDF](341.2KB)
Abstract:
High robustness of complex ecological systems in the face of species extinction has been hypothesized based on the redundancy in species. We explored how differences in network topology may affect robustness. Ecological bipartite networks used to be small, asymmetric and sparse matrices. We created synthetic networks to study the influence of the properties of network dimensions asymmetry, connectance and type of degree distribution on network robustness. We used two extinction strategies: node extinction and link extinction, and three extinction sequences differing in the order of species removal (least-to-most connected, random, most-to-least connected). We assessed robustness to extinction of simulated networks, which differed in one of the three topological features. Simulated networks indicated that robustness decreases when (a) extinction involved those nodes belonging to the most species-rich guild and (b) networks had lower connectance. We also compared simulated networks with different degree- distribution networks, and they showed important differences in robustness depending on the extinction scenario. In the link extinction strategy, the robustness of synthetic networks was clearly determined by the asymmetry in the network dimensions, while the variation in connectance produced negligible differences.
2012, 7(3): 441-461 doi: 10.3934/nhm.2012.7.441 +[Abstract](3136) +[PDF](716.9KB)
Abstract:
Many distributed systems lend themselves to be modelled as networks, where nodes can have a range of attributes and properties based on which they may be classified. In this paper, we attempt the task of quantifying varying levels of similarity among nodes in a complex network over a period of time. We analyze how this similarity varies as nodes implement their functional logic and node states vary accordingly. We then use information theory to analyze how much Shannon information is conveyed by such a similarity measure, and how such information varies with time. We also propose node congruity as a measure to quantify the contribution of each node to the network's scalar assortativity. Finally, focussing on networks with binary states, we present algorithms (logic functions) which can be implemented in nodes to maximize or minimize scalar assortativity in a given network, and analyze the corresponding tendencies in information content.
2012, 7(3): 463-471 doi: 10.3934/nhm.2012.7.463 +[Abstract](2496) +[PDF](979.0KB)
Abstract:
We introduce the notions of centrality interference and centrality robustness, as measures of variation of centrality values when the structure of a network is modified by removing or adding individual nodes from/to a network. Centrality analysis allows categorizing nodes according to their topological relevance in a network. Thus, centrality interference analysis allows understanding which parts of a network are mostly influenced by a node and, conversely, centrality robustness allows quantifying the functional dependency of a node from other nodes in the network. We examine the theoretical significance of these measures and apply them to classify nodes in a road network to predict the effects on the traffic jam due to variations in the structure of the network. In these case the interference analysis allows to predict which are the distinct regions of the network affected by the function of different nodes. Such notions, when applied to a variety of different contexts, opens new perspectives in network analysis since they allow predicting the effects of local network modifications on single node as well as global network functionality.
2012, 7(3): 473-481 doi: 10.3934/nhm.2012.7.473 +[Abstract](2850) +[PDF](489.7KB)
Abstract:
Many diseases have a genetic origin, and a great effort is being made to detect the genes that are responsible for their insurgence. One of the most promising techniques is the analysis of genetic information through the use of complex networks theory. Yet, a practical problem of this approach is its computational cost, which scales as the square of the number of features included in the initial dataset. In this paper, we propose the use of an iterative feature selection strategy to identify reduced subsets of relevant features, and show an application to the analysis of congenital Obstructive Nephropathy. Results demonstrate that, besides achieving a drastic reduction of the computational cost, the topologies of the obtained networks still hold all the relevant information, and are thus able to fully characterize the severity of the disease.
2012, 7(3): 483-501 doi: 10.3934/nhm.2012.7.483 +[Abstract](2144) +[PDF](411.5KB)
Abstract:
The Dirichlet problem and Dirichlet to Neumann map are analyzed for elliptic equations on a large collection of infinite quantum graphs. For a dense set of continuous functions on the graph boundary, the Dirichlet to Neumann map has values in the Radon measures on the graph boundary.
2012, 7(3): 503-524 doi: 10.3934/nhm.2012.7.503 +[Abstract](2660) +[PDF](446.1KB)
Abstract:
In this work, we are concerned with the convergence of the multiscale finite element method (MsFEM) for elliptic homogenization problems, where we do not assume a certain periodic or stochastic structure, but an averaging assumption which in particular covers periodic and ergodic stochastic coefficients. We also give a result on the convergence in the case of an arbitrary coupling between grid size $H$ and a parameter $\epsilon$. $\epsilon$ is an indicator for the size of the fine scale which converges to zero. The findings of this work are based on the homogenization results obtained in [B. Schweizer and M. Veneroni, The needle problem approach to non-periodic homogenization, Netw. Heterog. Media, 6 (4), 2011].
2012, 7(3): 525-541 doi: 10.3934/nhm.2012.7.525 +[Abstract](2023) +[PDF](444.2KB)
Abstract:
We prove that the signed porous medium equation can be regarded as limit of an optimal transport variational scheme, therefore extending the classical result for positive solutions of [13] and showing that an optimal transport approach is suited even for treating signed densities.
2012, 7(3): 543-582 doi: 10.3934/nhm.2012.7.543 +[Abstract](2052) +[PDF](541.1KB)
Abstract:
This paper deals with the description of the overall effect of pinning conditions in discrete systems. We study a variational problem on the discrete in which pinning sites are modeled as network subsets on which concentrated forces are imposed. We want to determine the asymptotic effect of pinning conditions on a periodic lattice as its size vanishes. Our analysis is performed in the framework of $\Gamma$-convergence and highlights the analogies and differences with the corresponding continuous problem, i.e. periodically perforated domains. We derive a functional form for the limit energies which depends on the relationship between the space dimension and the growth rate of the interaction functions.

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