Airports demo

demo/airports.m

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+%% The US airport network
+% THe North American airport network is an interesting graph to examine.
+% The source for this data was a file on Brendan Frey's affinity
+% propagation website.  A(i,j) is the negative travel time between two
+% airports.  Although the data didn't include the airport locations, I used
+% the Yahoo! Geocoding API to generate a latitude and longitude for each
+% airport.
+
+%% The data
+load_gaimc_graph('airports');
+%% 
+% Plot a histogram for all route time estimates
+[si, ti, rt] = find(A);
+hist(-rt,100);  % times are stored as negative values
+
+%%
+% Find the lengthiest route
+[val,ind] = max(-rt)
+{labels{si(ind)} labels{ti(ind)}}
+
+%%
+% Some of the routes include stop overs, so it's probable that is what 
+% we find in this case.
+
+%% Graph analysis: connected?
+% One of the first questions about any graph should be if it's connected or
+% not. 
+
+max(scomponents(A))
+
+%%
+% There is only one connected component, so the graph is connected.
+
+%% Distance instead of time
+% Let's see how the edges correlate distance with estimated travel time.
+[ai aj te] = find(A);
+de = distance(xy(ai,:), xy(aj,:));
+plot(de,-te,'.');
+xlabel('distance (arclength)'); ylabel('time (?)');
+
+%%
+% Wow!  It's all over the place, but there is a lower bound.  Some of
+% these routes can include stop
+
+%% Minimum spanning tree 
+% This section repeats and extends some analysis in the overall gaimc demo.
+% First, let's recompute the minimum spanning tree based on travel time.
+load_gaimc_graph('airports')
+A = -A;          % we store the negative travel time
+A = max(A,A');   % travel time isn't symmetric
+T = mst_prim(A);
+clf;
+gplot(T,xy);
+
+
+% These next lines plot a map of the US with states colored.  
+ax = worldmap('USA');
+load coast
+geoshow(ax, lat, long,...
+    'DisplayType', 'polygon', 'FaceColor', [.45 .60 .30])
+states = shaperead('usastatelo', 'UseGeoCoords', true);
+faceColors = makesymbolspec('Polygon',...
+    {'INDEX', [1 numel(states)], 'FaceColor', polcmap(numel(states))});
+    geoshow(ax, states, 'DisplayType', 'polygon', 'SymbolSpec', faceColors)
+set(gcf,'Position', [   52   234   929   702]);
+%%
+% That's the US, now we need to plot our data on top of it.
+[X,Y] = gplot(T,xy); % get the information to reproduce a gplot
+plotm(Y,X,'k.-','LineWidth',1.5); % plot the lines on the map 
+%% 
+% We need to clear the axes after the mapping toolbox
+clf;
+%%
+% Let's just look at the continential US too.
+figure; ax = usamap('conus');
+states = shaperead('usastatelo', 'UseGeoCoords', true, 'Selector',...
+    {@(name) ~any(strcmp(name,{'Alaska','Hawaii'})), 'Name'});
+faceColors = makesymbolspec('Polygon',...
+    {'INDEX', [1 numel(states)], 'FaceColor', polcmap(numel(states))});
+geoshow(ax, states, 'DisplayType', 'polygon', 'SymbolSpec', faceColors)
+framem off; gridm off; mlabel off; plabel off
+set(gcf,'Position', [   52   234   929   702]);
+plotm(Y,X,'k.-','LineWidth',1.5); % plot the lines on the map 
+%%
+% One interesting aspect of this map is that major airline hubs (Chicago,
+% New York, etc. are not well represented.  One possible explaination is
+% that they have larger delays than other regional airports.
+%
+% Clear the figure again before proceeding.
+clf;
+
+%% Honolulu to St. Johns?
+% Before, we saw that the lengthiest route was between St. John's and
+% Honolulu.  But, does the network have a better path to follow between
+% these cities?  Let's check using Dijkstra's shortest path algorithm!
+
+% Reload the network to restore it.
+load_gaimc_graph('airports')
+A = -A;
+
+% find the longest route again
+[si, ti, rt] = find(A);
+[val,ind] = max(rt); % we've already negated above, so no need to redo it
+start = si(ind);
+dest = ti(ind);
+[d pred] = dijkstra(A,start); % compute the distance to everywhere from St. Johns
+d(dest)
+
+%%
+% That value is considerably shorter than the direct time.  How do we find
+% this awesome route?
+path =[]; u = dest; while (u ~= start) path=[u path]; u=pred(u); end
+fprintf('%s',labels{start}); 
+for i=path; fprintf(' --> %s', labels{i}); end, fprintf('\n');
+
+%% All pairs shortest paths
+% At this point, the right thing to do would be to recompute each edge in
+% the network using an all-pairs shortest path algorithm, and then look at
+% how distance correlates with time in that network.  However, I haven't
+% had time to implement that algorithm yet.
+
+%% Conclusion
+% Hopefully, you will agree that network algorithms are a powerful way to
+% look at the relationships between airports!