Limited resolution and multiresolution methods in complex network community detection

Detecting community structure in real-world networks is a challenging problem. Recently, it has been shown that the resolution of methods based on optimizing a modularity measure or a corresponding energy is limited; communities with sizes below some threshold remain unresolved. One possibility to go around this problem is to vary the threshold by using a tuning parameter, and investigate the community structure at variable resolutions. Here, we analyze the resolution limit and multiresolution behavior for two different methods: a q-state Potts method proposed by Reichard and Bornholdt, and a recent multiresolution method by Arenas, Fernandez, and Gomez. These methods are studied analytically, and applied to three test networks using simulated annealing.Comment: 6 pages, 2 figures.Minor changes from previous version, shortened a couple of page

A powder metallurgy austenitic stainless steel for application at very low temperatures

The Large Hadron Collider to be built at CERN will require 1232 superconducting dipole magnets operating at 1.9 K. By virtue of their mechanical properties, weldability and improved austenite stability, nitrogen enriched austenitic stainless steels have been chosen as the material for several of the structural components of these magnets. Powder Metallurgy (PM) could represent an attractive production technique for components of complex shape for which dimension tolerances, dimensional stability, weldability are key issues during fabrication, and mechanical properties, ductility and leak tightness have to be guaranteed during operation. PM Hot Isostatic Pressed test plates and prototype components of 316LN-type grade have been produced by Santasalo Powdermet Oy. They have been fully characterized and mechanically tested down to 4.2 K at CERN. The fine grained structure, the absence of residual stresses, the full isotropy of mechanical properties associated to the low level of Prior Particle Boundaries oxides resulted in superior mechanical properties and high ductility down to liquid helium temperature. The ready weldability and the leak tightness of the alloy have been demonstrated. The properties measured on test plates are comparable to those found in real components, such as prototype end covers fabricated by the same PM technique

Emergence of communities in weighted networks

Topology and weights are closely related in weighted complex networks and this is reflected in their modular structure. We present a simple network model where the weights are generated dynamically and they shape the developing topology. By tuning a model parameter governing the importance of weights, the resulting networks undergo a gradual structural transition from a module free topology to one with communities. The model also reproduces many features of large social networks, including the "weak links" property.Comment: 4 pages, 5 figure

Large-scale structure of a nation-wide production network

Production in an economy is a set of firms' activities as suppliers and customers; a firm buys goods from other firms, puts value added and sells products to others in a giant network of production. Empirical study is lacking despite the fact that the structure of the production network is important to understand and make models for many aspects of dynamics in economy. We study a nation-wide production network comprising a million firms and millions of supplier-customer links by using recent statistical methods developed in physics. We show in the empirical analysis scale-free degree distribution, disassortativity, correlation of degree to firm-size, and community structure having sectoral and regional modules. Since suppliers usually provide credit to their customers, who supply it to theirs in turn, each link is actually a creditor-debtor relationship. We also study chains of failures or bankruptcies that take place along those links in the network, and corresponding avalanche-size distribution.Comment: 17 pages with 8 figures; revised section VI and references adde

A shadowing problem in the detection of overlapping communities: lifting the resolution limit through a cascading procedure

Community detection is the process of assigning nodes and links in significant communities (e.g. clusters, function modules) and its development has led to a better understanding of complex networks. When applied to sizable networks, we argue that most detection algorithms correctly identify prominent communities, but fail to do so across multiple scales. As a result, a significant fraction of the network is left uncharted. We show that this problem stems from larger or denser communities overshadowing smaller or sparser ones, and that this effect accounts for most of the undetected communities and unassigned links. We propose a generic cascading approach to community detection that circumvents the problem. Using real and artificial network datasets with three widely used community detection algorithms, we show how a simple cascading procedure allows for the detection of the missing communities. This work highlights a new detection limit of community structure, and we hope that our approach can inspire better community detection algorithms.Comment: 14 pages, 12 figures + supporting information (5 pages, 6 tables, 3 figures

Detecting modules in dense weighted networks with the Potts method

We address the problem of multiresolution module detection in dense weighted networks, where the modular structure is encoded in the weights rather than topology. We discuss a weighted version of the q-state Potts method, which was originally introduced by Reichardt and Bornholdt. This weighted method can be directly applied to dense networks. We discuss the dependence of the resolution of the method on its tuning parameter and network properties, using sparse and dense weighted networks with built-in modules as example cases. Finally, we apply the method to data on stock price correlations, and show that the resulting modules correspond well to known structural properties of this correlation network.Comment: 14 pages, 6 figures. v2: 1 figure added, 1 reference added, minor changes. v3: 3 references added, minor change

The Naming Game in Social Networks: Community Formation and Consensus Engineering

We study the dynamics of the Naming Game [Baronchelli et al., (2006) J. Stat. Mech.: Theory Exp. P06014] in empirical social networks. This stylized agent-based model captures essential features of agreement dynamics in a network of autonomous agents, corresponding to the development of shared classification schemes in a network of artificial agents or opinion spreading and social dynamics in social networks. Our study focuses on the impact that communities in the underlying social graphs have on the outcome of the agreement process. We find that networks with strong community structure hinder the system from reaching global agreement; the evolution of the Naming Game in these networks maintains clusters of coexisting opinions indefinitely. Further, we investigate agent-based network strategies to facilitate convergence to global consensus.Comment: The original publication is available at http://www.springerlink.com/content/70370l311m1u0ng3

Router-level community structure of the Internet Autonomous Systems

The Internet is composed of routing devices connected between them and organized into independent administrative entities: the Autonomous Systems. The existence of different types of Autonomous Systems (like large connectivity providers, Internet Service Providers or universities) together with geographical and economical constraints, turns the Internet into a complex modular and hierarchical network. This organization is reflected in many properties of the Internet topology, like its high degree of clustering and its robustness. In this work, we study the modular structure of the Internet router-level graph in order to assess to what extent the Autonomous Systems satisfy some of the known notions of community structure. We show that the modular structure of the Internet is much richer than what can be captured by the current community detection methods, which are severely affected by resolution limits and by the heterogeneity of the Autonomous Systems. Here we overcome this issue by using a multiresolution detection algorithm combined with a small sample of nodes. We also discuss recent work on community structure in the light of our results