# Symmetry in Complex Networks

## Abstract

**:**

## 1. Some Previous Concepts

In any graph, the sum of the degrees of all nodes (or “total degree”)

is equal to twice the number of edges.

In any graph there is an even number of nodes with an odd degree.

## 2. Symmetry and Networks

It is asymmetry that creates a phenomenon.

If an ensemble of causes is invariant with respect to any transformation,

the ensemble of their effects is invariant with respect to the same transformation.

The symmetry group of the cause is a subgroup of the symmetry group of the effect.

Also from Joe Rosen is the quote:The effect is at least as symmetric as the cause (and might be greater).

Recognized causal relations in nature are expressed as laws.

So,Laws impose equivalence relations in the state sets of causes and of effects.

Equivalent states of a cause are mapped to (i.e., correlated with)

This is the Equivalence Principle.equivalent states of its effect.

Concerning the Equivalence Principle for Processes on isolated physical systems, we can say that:Equivalent causes are associated with equivalent effects.

Equivalent initial states must evolve into equivalent states

And the General Symmetry Evolution Principle:(while inequivalent states may evolve into equivalent states).

This assertion can also be stated as:The “initial” symmetry group is a subgroup of the “final” symmetry group.

For an isolated physical system the degree of symmetry

cannot decrease as the system evolves; instead,

Finally, we have the Special Symmetry Evolution Principle:it either remains constant or increases.

As an isolated system evolves, the populations of the equivalence

classes of the sequence of states through which it passes cannot

Equivalently,decrease, but either remain constant or increase.

The degree of symmetry of the state of an isolated system

cannot decrease during evolution; instead, it either

remains constant or increases.

The degree of symmetry of a macrostate of stable equilibrium must be relatively high.

Ugly Symmetry-Beautiful Diversity

“Life is beautiful but full of asymmetry”

## 3. Symmetry as Invariance

## 4. Random Graphs

^{L}

## 5. Self-Similarity

## 6. Small-World Model

## 7. Scale-Free Networks

^{-γ}

## 8. Diameter of the Web

_{Web}> = 18.59

## 9. Community Structure

_{ij}), where a

_{ij}= 1, if nodes i and j are connected; otherwise, a

_{ij}= 0. Then, the modularity function, denoted by Q, will be defined as:

_{k}) = Σ [{L(V

_{j}, V

_{j})/L(V, V)} - {L(V

_{j}, V)/L(V, V)}

^{2}]

_{k}is a partition of the nodes into k groups, and:

_{i}

_{∈V´,i}

_{∈V´´}a

_{ij}

## 10. Fuzzy Symmetry

_{i}log p

_{i}

**R**

^{n}. And monotonically to −∞ along rays from the origin. So, such a minimum is always achieved and it will be finite.

- I)
- m (∅) = 0;
- II)
- m (U) =1;
- III)
- If A, B ∈ ℘, with A ⊆ B, then m (A) ≤ m (B) [monotonicity].

- I)
- If A is a crisp set, then H (A) = 0;
- II)
- If H (x) = 1/2, for each x∈A, then H (A) is maximal (total uncertainty);
- III)
- If A is less fuzzified than B, it holds that H (A) ≤ H (B);
- IV)
- H (A) = H (U\A).

^{U}→ [0,1]

- I)
- Sp (∅) = 0;
- II)
- Sp (k) = 1 if and only if k is a unitary set (singleton);
- III)
- If V and W are normal fuzzy sets in U, with V ⊂ W, then Sp (V) ≥ Sp (W);

^{U}denotes the class of fuzzy sets in U; Let (E, d) be a fuzzy metric space.

_{i}}

_{i}

_{∈{s,a}}

_{s}(A) = Sp(A) ((1-c(A))/(1+c(A)) + (1/(1+H(A))

_{a}(A ) = 1 - {Sp(A) ((1-c(A))/(1+c(A))+(1/(1+H(A))}

## 11. New Lines of Research

- i)
- The first law of information theory. The logarithmic function L = ln w, or the sum of entropy and information, L = S + I, of an isolated system remains unchanged, where S denotes the entropy and I the information content of the system.
- ii)
- The second law of information theory. Information of an isolated system decreases to a minimum at equilibrium.
- iii)
- The third law of information theory. For a perfect crystal (at zero absolute thermodynamic temperature), the information is zero and the static entropy is at the maximum. Or in a more general form, “for a perfect symmetric static structure, the information is zero and the static entropy is the maximum”.

- iv)
- Similarity principle. The higher the similarity among the components is, the higher the value of entropy will be and the higher the stability will be.

## 12. Conclusions

## Acknowledgments

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Garrido, A.
Symmetry in Complex Networks. *Symmetry* **2011**, *3*, 1-15.
https://doi.org/10.3390/sym3010001

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Garrido A.
Symmetry in Complex Networks. *Symmetry*. 2011; 3(1):1-15.
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Garrido, Angel.
2011. "Symmetry in Complex Networks" *Symmetry* 3, no. 1: 1-15.
https://doi.org/10.3390/sym3010001