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We introduce the (private) entropy of a directed graph (in a new networkcoding sense) as well as a number of related concepts. We show that the entropyof a directed graph is identical to its guessing number and can be bounded frombelow with the number of vertices minus the size of the graph's shortest indexcode. We show that the Network Coding solvability of each specific multipleunicast network is completely determined by the entropy (as well as by theshortest index code) of the directed graph that occur by identifying eachsource node with each corresponding target node.  Shannon's information inequalities can be used to calculate upper bounds on agraph's entropy as well as calculating the size of the minimal index code.Recently, a number of new families of so-called non-shannon-type informationinequalities have been discovered. It has been shown that there existcommunication networks with a capacity strictly less than required forsolvability, but where this fact cannot be derived using Shannon's classicalinformation inequalities. Based on this result we show that there exist graphswith an entropy that cannot be calculated using only Shannon's classicalinformation inequalities, and show that better estimate can be obtained by useof certain non-shannon-type information inequalities.

Graph Entropy, Network Coding and Guessing games