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

Effect of the telescopic process on <i>E</i>_<i>glob</i>.

<p>Effect of the telescopic process on subways (leftmost column), the US airline and city-based online social networks (rightmost column) as a function of <i>f</i>. The statistical properties considered in these panels are topological and metrical <i>E</i>_<i>glob</i>. The abstraction process does not preserves the topological <i>E</i>_<i>glob</i> (top panels) while varying <i>f</i>. In particular, regardless of the network considered, the networks viewed at macro level are simpler and more efficient compared to micro view. Conversely, the situation is slightly different for metrical <i>E</i>_<i>glob</i> (bottom panels). In this case, the connection pattern of the system considered alters significantly the outcome of the abstraction process. In fact, we detected that the structure of subway networks allow a good preservation of the metrical efficiency in the spectrum whereas in city-based online social networks this feature is absent.</p

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

Graph types.

<p>Examples of different graph types.</p

Sequence of box covering iteration for telescopic analysis.

<p>Example of grids applied on top of networks as a function of fuzziness. Leftmost grid has low fuzziness <i>f</i> = 0.125 whereas the rightmost has <i>f</i> = 1. The granularity of the spectrum in this example is equal to 7. In this paper, we only consider linear increase of <i>f</i>.</p

One-step abstraction process.

<p>One-step application of the abstraction process to a small graph. (a) Original graph <i>G</i>. Red (dashed) circles identify the group of nodes that will be merged together. (b) Output graph <i>G</i>_<i>i</i> in which nodes <i>c</i>, <i>f</i>, <i>e</i>, <i>h</i>, <i>l</i> and <i>n</i>, <i>m</i> are collapsed into new nodes <i>e</i>, <i>c</i>, <i>h</i> ∈ <i>V</i>_<i>i</i> respectively. Coordinates are the barycenter of collapsed nodes. Three edges are then removed because they connect the collapsed nodes: (<i>n</i>, <i>m</i>), (<i>c</i>, <i>e</i>), and (<i>f</i>, <i>e</i>).</p

Effect of the telescopic abstraction on the diameter as a function of <i>f</i>.

<p>Effect of the telescopic abstraction for the physical <i>D</i>_<i>p</i>, topological <i>D</i>_<i>t</i> and metrical <i>D</i>_<i>m</i> diameter as a function of fuzziness <i>f</i>. All the values were normalized by the baseline values at <i>f</i> = 0 (i.e., no abstraction is applied). The top panels contain results of subways, the bottom ones of city-based online social networks and the US airline network.</p

<i>P</i>_<i>cum</i> distribution of subways, transportation and social networks.

<p>The log-log plots of the cumulative degree distributions <i>P</i>_<i>cum</i>(<i>k</i>) of subways (Boston, Milan, New York, Paris, a to d), the US airline (e) and city-based online social networks (letter f to j) of Italy, Australia, The Netherlands, India and the United Kingdom. The distributions are characterized by exponents <i>γ</i> of <i>P</i>(<i>k</i>) ∼ <i>k</i>^−<i>γ</i> that is one plus the slope of <i>P</i>_<i>cum</i>(<i>k</i>) (in a log-log plot), i.e. <i>γ</i> = 1 + <i>γ</i>_<i>cum</i>. The coefficient is <i>γ</i> = 3.5 for subways networks, 2.6 for the US airline, 1.85 for Indian city-based online social network, 1.68 for the United Kingdom, 2.61 for Italy, 1.94 for Australia and 1.61 for the Netherlands. The coefficients for subways might not be precise due to the small dimension of the networks.</p

Example of micro-macro analysis.

<p>Example of micro-macro analysis obtained by increasing (abstraction process to the macro world) fuzziness <i>f</i>. When <i>f</i> = 0, no abstraction is applied whereas at increasing values of <i>f</i>, the network will be more obfuscated and the structure will be simpler. In the extreme situation when <i>f</i> is maximum, <i>f</i> = 1 (not displayed in the figure), the original network will be collapsed into a one node graph.</p