Análisis de las caricaturas de los diarios El Universo y El Telégrafo

In order to state our axioms, we need the following definitions. In Figure 8.2, we give a visualization of the quantities appearing in these definitions.

Definition 8.3. Given a graphG = (V, E), if s ∈ V , we denote by Γ`(s) the set of nodes at distance exactly` from s, and we let γ`<sub>(s) = |Γ</sub>`

(s)|. Similarly, we denote by N`(s) the set of nodes at distance at most` from s, and we let n`<sub>(s) = |N</sub>`

(s)|.

Definition 8.4. Let x be a real number such that 0 < x < 1: we denote by τs(nx) =

min{` ∈ N : γ`_{(s) > n}x_{}, and by T (d → n}x_{) the average number of steps for a node of degree}

d to obtain a neighborhood of nx_{nodes (more formally,T (d → n}x_{) is the average of τ} s(nx)

8.2 - The Axioms 149 s γ1_(s) Γ1(s) Γ2(s) γ 2_(s) ... First time > nx τ s (n x ) N1(s) N2(s)

Figure 8.2. A visualization of the quantities appearing in Definitions 8.3 and 8.4. The green triangle represents the tree obtained by performing a BFS from s, stopped as soon as we hit a neighbor with size at least nx_.

Remark 8.5. In general, τs(nx) could not be defined, if no neighborhood of s has size at

leastnx_{. In random graphs, we solve this issue by showing that τ}

s(nx) is defined for each

node in the giant component, because it containsΘ(n) nodes and has diameter O(log n). In real-world graphs, since the diameter is usually very small, τs(nx) is defined for each value

ofx considered in our experiments.

Our axioms depend on a parameterε: for instance, the first axiom bounds the number of nodess such that τs(nx) ≥ (1 + ε)T (d → nx). Intuitively, one can think of ε as a constant

which is smaller than any other constant appearing in the proofs, but bigger than _n1, or any other infinitesimal function of n. Indeed, in random graphs, we prove that if we fix ε, δ > 0, we can find nε,δ such that the axioms hold for each n > nε,δ, with probability at

least1 − δ. In real-world graphs, we experimentally show that the axioms are satisfied with good approximation forε = 0.2. In our analyses, the time bounds are of the form nc+O(ε)_,

and the constants in the O are quite small. Since, in our dataset,n0.2 _{is between6 and 19,}

we can safely considernc+O(ε) _{close ton}c_.

The first axiom analyzes the typical and extremal values of τs(nx), where s is any node.

Axiom 1. There exists a constantc such that: • for each node s with degree d > nε_{, τ}

s(nx) ≤ (1 + ε) T (d → nx) + 1
;

• the number of nodes satisfying τs(nx) ≥ (1 + ε) T (d → nx) + α
is O ncα−x
;

• the number of nodes satisfying τs(nx) ≥ (1 − ε) T (1 → nx) + α
is Ω ncα−x
.

In random graphs, the values ofT (d → nx_{) depend on the exponent β (see Table 8.2). In}

many of our analyses, we do not use the actual values ofT (d → nx_{), but we use the following}

properties:

• T d → nx+ε
_{≤ T (d → n}x_{) (1 + O(ε));}

• P∞

d=1|{v ∈ V : deg(v) = d}|T (d → nx) = (1 + o(1))nT (1 → nx);

• T (1 → nx_{) + T 1 → n}1−x
_{− 1 = (1 + o(1)) dist}

avg(n), where distavg(n) is a function

not depending on x (this function is very close to the average distance distavg, as we

Table 8.2. The values of T (d → nx), distavg(n) and c, depending on the value of β. All these values should

be multiplied by (1 + o(1)), which is omitted to increase readability.

Regime T (d → nx_{) dist} avg(n) c 1 < β < 2 1 if d ≥ nx_{, 2 otherwise 3 n}−2−β β−1+o(1) 2 < β < 3 log 1 β−2 log nx log d ifn x_{< n}β−11 2 log ₁ β−2log n η(1) log 1 β−2 log nx log d + O(1) if n x_{> n}β−11 β > 3 logM1(µ) nx d logM1(µ)n η(1)

The other two axioms relate the distance between two nodess, t with the values of τs(nx),

τt(ny), where x, y are two reals between 0 and 1. The idea behind these axioms is to apply

the “birthday paradox”, assuming that Γτs(nx)_{(s) and Γ}τt(ny)_{(t) are random sets of n}x _and

ny _{nodes. In this idealized setting, ifx + y > 1, there is a node that is common to both, and}

dist(s, t) ≤ τs(nx) + τt(ny); conversely, if x + y < 1, dist(s, t) is likely to be bigger than

τs(nx) + τt(ny). Let us start with the simplest axiom, which deals with the case x + y > 1.

Axiom 2. Let us fix two real numbers0 < x, y < 1 such that x + y > 1 + ε. For each pair of nodess, t, dist(s, t) < τs(nx) + τt(ny).

The third axiom is a sort of converse: the main idea is that, if the product of the size of two neighborhoods is smaller thann, then the two neighborhoods are usually not connected. The simplest way to formalize this is to state that, for each pair of nodess, t, dist(s, t) ≥ τs(nx) +

τt(ny). However, there are two problems with this statement: first, in random graphs, if we

fixs and t, dist(s, t) ≥ τs(nx)+τt(ny) a.a.s., not w.h.p., and hence there might be o(n) nodes

t such that dist(s, t) < τs(nx) + τt(ny) (for example, if s and t are neighbors, they do not

satisfydist(s, t) ≥ τs(nx) + τt(ny)). To solve this, our theorem bounds the number of nodes

t satisfying dist(s, t) ≥ τs(nx)+τt(ny). The second problem is more subtle: for example, if s

has degree1, and its only neighbor has degree n12, τ_s  n14  = τs  n12 

= 2, and the previous statement cannot hold forx = 1₄. However, this problem does not occur ifx ≥ y: the intuitive idea is that we can “ignore” nodes with degree bigger thannx_{. Indeed, if a shortest path from}

s to t passes through a node v with degree bigger than nx_{, then τ}

s(nx) ≤ dist(s, v) + 1,

τt(ny) ≤ dist(t, v) + 1, and hence dist(s, t) = dist(s, v) + dist(v, t) ≥ τs(nx) + τt(ny) − 2.

Axiom 3. Let0 < z ≤ y < x < 1, let x + y ≥ 1 + ε, and let α, ω be integers. If Tα,ω,z is the

set of nodes t such that τt(nz) is between α and ω, there are at most |Tα,ω,z|n

x+y+ε

n nodes

t ∈ T such that dist(s, t) < τs(nx) + τt(ny) − 2.

Finally, in some analyses, we also need to use the fact that the degree distribution is power law. To this purpose, we add a further axiom (in random graphs, this result is well-known [159, 160]).

Axiom 4. The number of nodes with degree bigger thand is Θ n

dmax(1,β−1) 

.