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

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

The Netherlands’s city-based online social network.

<p>The online social networks of the Netherlands 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

Graph types.

<p>Examples of different graph types.</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

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

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

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

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