Micro-Macro Analysis of Complex Networks

Complex systems have attracted considerable interest because of their wide range of applications, and are often studied via a \u201cclassic\u201d approach: study a specific system, find a complex network behind it, and analyze the corresponding properties. This simple methodology has produced a great deal of interesting results, but relies on an often implicit underlying assumption: the level of detail on which the system is observed. However, in many situations, physical or abstract, the level of detail can be one out of many, and might also depend on intrinsic limitations in viewing the data with a different level of abstraction or precision. So, a fundamental question arises: do properties of a network depend on its level of observability, or are they invariant? If there is a dependence, then an apparently correct network modeling could in fact just be a bad approximation of the true behavior of a complex system. In order to answer this question, we propose a novel micro-macro analysis of complex systems that quantitatively describes how the structure of complex networks varies as a function of the detail level. To this extent, we have developed a new telescopic algorithm that abstracts from the local properties of a system and reconstructs the original structure according to a fuzziness level. This way we can study what happens when passing from a fine level of detail (\u201cmicro\u201d) to a different scale level (\u201cmacro\u201d), and analyze the corresponding behavior in this transition, obtaining a deeper spectrum analysis. The obtained results show that many important properties are not universally invariant with respect to the level of detail, but instead strongly depend on the specific level on which a network is observed. Therefore, caution should be taken in every situation where a complex network is considered, if its context allows for different levels of observability

Multidimensional analysis of complex networks

Complex Networks analysis turn out to be a very promising field of research, testified by many research projects and works that span different fields. Those analysis have been usually focused on characterize a single aspect of the system and a study that considers many informative axes along with a network evolve is lacking. We propose a new multidimensional analysis that is able to inspect networks in the two most important dimensions, space and time. To achieve this goal, we studied them singularly and investigated how the variation of the constituting parameters drives changes to the network as a whole. By focusing on space dimension, we characterized spatial alteration in terms of abstraction levels. We proposed a novel algorithm that, by applying a fuzziness function, can reconstruct networks under different level of details. We verified that statistical indicators depend strongly on the granularity with which a system is described and on the class of networks. We keep fixed the space axes and we isolated the dynamics behind networks evolution process. We detected new instincts that trigger social networks utilization and spread the adoption of novel communities. We formalized this enhanced social network evolution by adopting special nodes (called sirens) that, thanks to their ability to attract new links, were able to construct efficient connection patterns. We simulated the dynamics of the system by considering three well-known growth models. Applying this framework to real and synthetic networks, we showed that the sirens, even when used for a limited time span, effectively shrink the time needed to get a network in mature state. In order to provide a concrete context of our findings, we formalized the cost of setting up such enhancement and provided the best combinations of system's parameters, such as number of sirens, time span of utilization and attractiveness

Strategies of Success for Social Networks: Mermaids and Temporal Evolution

The main goal of this article is to investigate techniques that can quickly lead to successful social systems by boosting network connectivity. This is especially useful when starting new online communities where the aim is to increase the system utilization as much as possible. This aspect is very important nowadays, given the existence of many online social networks available on the web, and the relatively high level of competition. In other words, attracting users’ attention is becoming a major concern, and time is an essential factor when investing money and resources into online social systems. Our study describes an effective technique that deals with this issue by introducing the notion of mermaids, special attractors that alter the normal evolutive behavior of a social system. We analyze how mermaids can boost social networks, and then provide estimations of fundamental parameters that business strategists can take into account in order to obtain successful systems within a constrained budget

United Kingdom’s city-based online social network.

<p>The online social networks of the United Kingdom created from its VirtualTourist online community. Lines (yellow) represent edges of the network connecting cities that share at least one friend. Background satellite image TIROS-3 courtesy of NASA (the U.S. National Aeronautics and Space Administration) and NOAA (the U.S. National Oceanic and Atmospheric Administration).</p

Box covering issue.

<p>Grid displacement issue when the distance between two nodes is less than fuzziness value. Wrong (a) and correct (b) grid displacement.</p

Statistical features of transportation and city-based online social networks.

<p>Datasets statistics of subways, the US airline and city-based online social networks: number of nodes <i>n</i> and edges <i>m</i> of the graphs, maximum degree <i>k</i>_<i>max</i> and average node degree ⟨<i>k</i>⟩, standard deviation of the degree <i>σ</i>_<i>k</i>, assortativity mixing by degree <i>ρ</i>, physical, topological and metrical diameter <i>D</i>, global and local efficiency <i>E</i>_<i>glob</i>, <i>E</i>_<i>loc</i>, costs and <i>C</i>/<i>E</i> property (defined as the ratio between cost and global efficiency). Both <i>topological</i> and <i>metrical</i> versions are calculated of the latter three indicators.</p><p>Statistical features of transportation and city-based online social networks.</p

Italy’s city-based online social network.

<p>The online social networks of Italy created from its VirtualTourist online community. Lines (yellow) represent edges of the network connecting cities that share at least one friend. Background satellite image TIROS-3 courtesy of NASA (the U.S. National Aeronautics and Space Administration) and NOAA (the U.S. National Oceanic and Atmospheric Administration).</p

Number of collapsed nodes and edges as a function of <i>f</i> in log-log axes.

<p>Number of collapsed nodes <i>n</i> and edges <i>m</i> as a function of <i>f</i> in log-log axes. The values are normalized by the baseline values <i>n</i>(0) and <i>m</i>(0) respectively, obtained at <i>f</i> = 0 (i.e., no abstraction applied). The leftmost panels refer to subway networks whereas the rightmost refer to city-based online social networks and the US airline network. The decrease of <i>n</i> and <i>m</i> is clearly exponential, even though the rate is influenced by many factors like network size and node positions.</p

Australia’s city-based online social network.

<p>The online social networks of Australia created from its VirtualTourist online community. Lines (yellow) represent edges of the network connecting cities that share at least one friend. Background satellite image TIROS-3 courtesy of NASA (the U.S. National Aeronautics and Space Administration) and NOAA (the U.S. National Oceanic and Atmospheric Administration).</p

Effect of the telescopic analysis on <i>c</i>_<i>t</i> and <i>c</i>_<i>m</i>.

<p>Effect of the telescopic analysis on topological and metrical cost (<i>c</i>_<i>t</i> and <i>c</i>_<i>m</i>) as a function of <i>f</i> for subways (leftmost column), the US airline and city-based online social networks (rightmost column). We note that our coarse graining process produces networks more expensive than detailed ones. This effect might be caused by the creation of redundant structures in macro level systems so that the whole cost will be higher. Even though both curves are positively correlated to <i>f</i>, the slope in subways networks is smaller compared to city-based online social networks. To verify whether this effect is not trivially caused by a low efficiency value, we will consider <i>C</i>/<i>E</i>_<i>glob</i> index (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0116670#pone.0116670.g020" target="_blank">Fig. 20</a>).</p