For mutualistic and gene regulatory networks, nonlinear tipping-point control importance favors high-degree nodes, while linear controllability importance favors low-degree nodes, indicating a systematic mismatch.
Maximum matching on random graphs
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abstract
The maximum matching problem on random graphs is studied analytically by the cavity method of statistical physics. When the average vertex degree \mth{c} is larger than \mth{2.7183}, groups of max-matching patterns which differ greatly from each other {\em gradually} emerge. An analytical expression for the max-matching size is also obtained, which agrees well with computer simulations. Discussion is made on this {\em continuous} glassy phase transition and the absence of such a glassy phase in the related minimum vertex covering problem.
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Irrelevance of linear controllability to nonlinear dynamical networks
For mutualistic and gene regulatory networks, nonlinear tipping-point control importance favors high-degree nodes, while linear controllability importance favors low-degree nodes, indicating a systematic mismatch.