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2) with ✦ representing the weighted real valued adjacency matrix, thus the entries of ✦ reflect the number of messages which go from vertex ✰ to vertex ✱ . 1) a sample communication can be seen. 1). The diagonal line has been drawn to demonstrate the underlying symmetry. 1) that can be transformed into a Hermitian matrix by rotation. 15). 3). 3) Proof : 1. Let the entry of ➶ and phase . ✗ be any complex number ✄✆☎✞✝ ❅→ï❃✜❵❫➸➵ with absolute value ï 2. 1) ✄❯✝÷☎ ❅→● ✄❯☎➟✝ ❅➘●❱ï✑✜✑✾❯❫➸➵ 3. ✄❯☎➟☎ ❅➐✳✙❅➐Ñ ☎✞☎ because of the exclusion of self references 4.
It explains ✗ ✗ ❢ ✗ to what extent a vector is stretched by or shrunk by ✾ if is nonsingular. 52). This norm will be the one used as the standard matrix norm, if not otherwise stated. ✫ ✗ If ✗ is nonsingular then ✫ ✗ ✾ ❢ ✫ ❅ ✫ ☞ ❅ ✤ ⑩✩ð➉ñ ✤ ❢ ò ó✈ô↕õ÷ö❃ò . 4. 46) Hermitian Operators in Hilbert Space A linear operator on a given field is defined as a function that sends every vector ✠ ✛✥ú into another vector ✦ ✛✥ú by a linear transformation. 47) ❛ ❜ ✦ ❫ ❫ ❫ with ❛ ☎✞✝ ✛Ïû can also be written as ✠ ❅❥û ✦ with ❛ ☎✞✝ the entries of the matrix û .
Excentricity ✜ ✪ being defined as ✜ ✪➓☎ ❅→➔ ✄✆✪➓✝ ✷✆●r❴❵st✵✸✰ ✱❲✹ . 8) ✒ which expresses the fact that this vertex has the minimal longest distance ✷✆●r❴❵st✵❱✰ ✱✸✹ of all vertices in the network. This measure is defined for both non-directional and directional settings. 202] prestige in comparison to centrality represents the degree to which a vertex is chosen by other vertices within a directed and possibly weighted network. 9). 9) ✵❲▼❧❦♠✲♥✹ The strength in this case is defined analogous to the the outbound strength.