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548], [Kat95, p. 53]). 42). 57). The eigenvalues can be interpreted as weights of the corresponding eigenspace. 57) with ✝✶☎ ❅ ✠☛☎q✠ ☎➦ ❅ ✟✠ ✪✡❢ ✪✡❢ ➛❁➛❁➛ ✪ ❜ ✪✡❢ ➛❁➛❵➛ ➛❁➛❵➛ ➛❁➛❵➛ ✪✡❢ ✪ ❜ ➛❁➛❁➛ ✪ ❜ ✪ ❜☛☞✡ representing the orthogonal projectors. Note, that ♦ ✝ ☎ ➧ ❅ ✝✶☎ . 5. 58) , ✝✢☎ ➦ ❅ ✝➌☎ , ✯ Any function ✌✐✵ ✬ ✹ of a Hermitian matrix ✬ ß can also Ý be seen as a function ✬ ☞ Ý ❞✎✍ ✬ ß Ý of its eigenvalues ✌✐✵ ✹ . Now let ✌✐✵ ✹➜❅ be a power series, then ❢ ✬ ✒ ♦ ☞ Ý ❞✎✍ ☞ ✒➢Ö it follows that ✌✐✵ ✹❳❅ ●þ❅ ✲ ➛❁➛❁➛❰▼ , since ✌✐✵ ✹❳❅✏✌✐✵❼Ò Ó✖Ò◗✾ ✹ ❫ ❫ ♦ with Ò representing the matrix of all eigenvectors and Ó a diagonal matrix ☞ with ✌✐✵ ✹ as entries.

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. After rotation about ❇ : Ñ ✞☎ ✝ ❅ 5. Ñ ÷✝ ☎ ❅ ✄✆☎✞✝ ✜❵❫✽❈◆❅→ï✑✜❵❫➸➵➫✜❵❫✽❈◆❅→ï✑✜❵❫❐❽❡➵ ❋ ❿ ✆✄ ✝÷☎ ✜❵❫✽❈✖❅➘●✸ï❃✜✑✾❯❫❸➵✑✜❵❫✘❈✖❅➐✜ ❫❃❂ ï❃✜❵❫❐❉❽ ❈➩✾✺➵ ❮ ➟☎ ✝ 6. For Hermitian ❿ matrices Ñ ❅ ï❃✜✑✾❯❫❐❽❁❂ ❮ ❋ ❈✺✾✺➵ . 20), therefore, ï✑✜❵❫❐❽❸➵ ❋ ❿ ❈ ❅ 44 CHAPTER 4 7. This holds, if ✿ ➶ ➑ ➶ ❇ ❅❨❦ ➧ ❦❊❇ ➑ .

If the star is being perturbed, the symmetry breaks down. This means that if, for example, a group shows the communication behavior of a star graph, additional cross communication takes the role of a perturbation as long as it is small compared to communication in the star. This is analog in the case of a complete graph. In this case for example one sided communication or very little communication between two nodes takes the form of a perturbation. 2 Eigenvectors and orthogonal projectors The absolute value of the eigenvector components is the well known eigenvector centrality index (see Bonacich and Lloyd [BL01]).

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