Hausdorff dimension obtained as the average from 150 paths in 10 proteins. 

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... the path is partitioned into N k segments, each r k in length. For the shortest path, we identify the length of segments r k with the cutoff distance r C and N k with k ij ,min . In Figure 3, we present results of calculations on 150 paths from the set of ten proteins. The straight line is the least squares fit to the data points. Its slope, which approximates the Hausdorff dimension given by Eq. 2 is D H = 1.1 , which is approximately the fractal dimension obtained in the preceding section. The maximum abscissa value of -1.9 corresponds to a cutoff radius of 6.8 Å and the minimum ordinate value of -2.4 corresponds to 11 Å. The points outside this range are not presented in the figure because below 6.8 Å, the jumps shown in Figure 2 lead to discontinuous data, and above 11 Å the system reaches a saturation level [13] and the slope equates to unity. Given the end residues, i { 1 } 0 and i { 1 } N . The numbers in parenthesis give the step number along the directed walk from the starting to the ending residue and the subscript indices in braces give the set of residues involved at each step. The set { j } at an intermediate point in general contains more than one residue, each of which are identified in the trip along the forward direction. The residues involved at each step are candidate residues that contain the residue of the shortest path which will be identified during backtracking (See below). In the first step, starting from i { 1 } ( 0 ) we find the set of residues i { j } ( 1 ) that can be accessed from i { 1 } ( 0 ) . The residues i { j } ( k ) are accessed from the residues of the previous step i { j } ( k − 1 ) . Residue j at the kth step will be accessed by ν { j } ( k ) different paths from residues of the previous step. The set of resides involved in going from i { 1 } ( 0 ) to i { 1 } ( N ) is { i { 1 } ( 0 ) , i { j } ( 1 ) , i { j } ( 2 ) ,  , i { j } ( k − 1 ) , i { j } ( N ) } and the number of arrivals to the residues from previous paths during the forward trip is { 1, { j } ( 1 ) , ν { j } ( 2 ) ,  , ν { j } ( k − 1 ) , ν { j } ( N ) } . The residues of the shortest path, { i 0 , i 1 , i 2 ,  , i k − 1 , i k ,  , i N } are obtained by backtracking, starting from N and going step by step to 1. When the residue at step k is identified in this way, the residue at step k-1 is determined from the list of residues that are adjacent to residue k, already determined in the previous step. This process of backtracking may lead to more than one shortest path, all of which may be identified individually by the backtracking method. In Figure 4, we present an example for the allosteric shortest path for 1GPW.pdb from residue LYS132 to ASP98. The shortest path with a cutoff radius of 6.9 Å reaches ASP98 in six steps. The dark vertical lines indicate the residues visited in each step during forward tracking, i.e., in going from LYS132 to ASP98. These are the residues that have to be visited while going from the initial to the final point of the path. Thus, there will be several paths from LYS132 to ASP98, but only one set of residues gives the shortest path, to be identified during backtracking. The dark circles are the residues of the path identified during backtracking. The importance of these residues in allosteric communication has been shown before [14, 15]. The path residues are situated in the highly visited regions in the figure. The list of residues visited during forward tracking is given in the Supplementary material section and the residues on the shortest path are highlighted. Identification of the shortest path residues also leads to the set 1, ν 1 , ν 2 ,  , ν k − 1 , ν k ,  , ν N identifying the number of visits to each residue along forward tracking, i.e., along the traveling direction of information. The Shannon entropy of the path is then defined as S / k = − f log f ...