Proof of Proposition 1

Notice that decay centrality can be rewritten as DCi(δ)=δDi+∑l=2n−1δlDil and consider the limit limδ→0DCi(δ)−DCj(δ)δ=00Di−Dj>0, where the last inequality is implied by Di>Dj. The expression that appears inside the limit is continuous in δ, therefore exists δ_ such that for all δ∈(0,δ_) it holds that DCi(δ)−DCj(δ)δ>0 or equivalently DCi(δ)>DCj(δ).

Proof of Proposition 2

If Di>Dj then the result holds immediately by Proposition 1. Otherwise, let l~≥2 be the first instance such that Dil~>Djl~. In this case, the difference between decay centralities can be written as DCi(δ)−DCj(δ)=∑l=l~n−1δl(Dil−Djl), because all previous terms of the sum are equal to zero. Hence, limδ→0DCi(δ)−DCj(δ)δl~=00Dil~−Djl~>0 and by continuity with respect to δ, there is δ_ such that for all δ∈(0,δ_) it holds that DCi(δ)−DCj(δ)δl~>0 or equivalently DCi(δ)>DCj(δ).

Proof of Proposition 3

We need three direct observations to obtain the result. First, note that Ci>Cj⇔∑k≠id(i,k)<∑k≠jd(j,k). Second, recall the alternative formulation of decay centrality as DCi(δ)=δDi+∑l=2n−1δlDil and third to observe that ∑k≠id(i,k)=Di+∑l=2n−1lDil. Therefore,

where the last inequality is implied by Ci>Cj. Hence, by continuity with respect to δ, exists δ¯ such that for all δ∈(δ¯,1) holds that DCi(δ)−DCj(δ)1−δ>0 or equivalently DCi(δ)>DCj(δ).

Proof of Proposition 4

If Ci>Cj the result holds immediately by Proposition 3. Otherwise, let l~≥2 be the first instance such that Cil~>Cjl~, hence also the first instance in which Fil~<Fjl~. Hence:

and by continuity with respect to δ, there is δ¯ such that for all δ∈(δ¯,1) it holds that DCi(δ)−DCj(δ)(1−δ)l~>0 or equivalently DCi(δ)>DCj(δ).

Proof of Proposition 5

DCi(δ)=δDi+∑l=2n−1δlDil, which gives DCi(1)=n−1 for all i. Therefore, the polynomial DCi(δ)−DCj(δ) has a root for δ=1 and another for δ=0, for any pair i,j∈N. Having observed that, we obtain the following expression.

where A1=Di−Dj and Al=Dil−Djl for l∈{2,…,n−1}, with the crucial observation being that ∑l=1n−1Al=0. This last equation allows us to rewrite the above expression as follows:

From the last expression is apparent that a sufficient condition, though not necessary, to satisfy DCi(δ)>DCj(δ) is that ∑l=1kAl≥0 for all k∈{1,…,n−1}, with at least one inequality being strict. This in turn is equivalent to ∑l=1kDil≥∑l=1kDjl for all k, which by definition means that Di>UDDj.

Proof of Proposition 6

Let ϵ=1−δ and restate decay centrality as follows: DCi(ϵ)=(1−ϵ)Di+∑l=2n−1(1−ϵ)lDil which gives DCi(ϵ=0)=n−1 for all i. Therefore, the polynomial DCi(ϵ)−DCj(ϵ) has a root for ϵ=1 (equivalent to δ=0) and another for ϵ=0 (equivalent to δ=1), for any pair i,j∈N. In addition to this, we consider the binomial identity: (1−ϵ)l=∑k=0l(−1)k(lk)ϵk, which allows us to rewrite DCi(ϵ)−DCj(ϵ) as follows:

where B1=Fi−Fj and Bl=Fil−Fjl for l∈{2,…,n−1}. Having observed that and knowing that ϵ=1 is also a root of the polynomial, we obtain the following expression via Euclidean division:

with the crucial observation being that ∑l=1n−1Bl=0, which is the remainder of the division of the polynomial by 1−ϵ, which we know that it should be equal to zero. Hence, similarly to Proposition 5, we obtain the following expression:

It is apparent from the last expression that a sufficient condition, though not necessary, to satisfy DCi(ϵ)>DCj(ϵ) for all ϵ∈(0,1), hence equivalently for all δ∈(0,1), is that ∑l=1kBl≤0 for all k∈{1,…,n−1}, with at least one inequality being strict. This in turn is equivalent to ∑l=1kFil≤∑l=1kFjl for all k, which by definition means that Fj>UDFi.

Proof of Proposition 7

1. DCi(δ)≥DCj(δ)⇒Diδ≥Djδ+[(n−1)−Dj]δ2⇒δ≤A1n−1−Dj. The last inequality holds for all δ∈(0,1/2] if 2A1≥(n−1)−Dj.

2.DCi(δ)≥DCj(δ)⇒Diδ+Di2δ2≥Djδ+Dj2δ2+[(n−1)−Dj−Dj2]δ3⇒δ≤A2+A22+4A1[(n−1)−Dj−Dj2]2[(n−1)−Dj−Dj2]. The last inequality holds for all δ∈(0,1/2] if A1+2A2≥(n−1)−Dj−Dj2.

3. The result can also be implied by Rouché’s Theorem, but it is presented here with an independent proof, which follows a similar process to the one used for obtaining Cauchy’s Bound of Polynomial Roots. More specifically, observe that by Proposition 1 together with A1>0 it must hold true that DCi(δ)−DCj(δ)=δ(A1+A2δ+⋯+An−1δn−2)>0 for δ close to zero and obviously has a root for δ=0. We also know that it has another root for δ=1. Then it is sufficient to show that DCi(δ)−DCj(δ) has no other root for δ∈(0,1/2) as long as A1≥|Al| for all l∈{2,…,n−1}. To prove this, it is sufficient to focus on the term (A1+A2δ+⋯+An−1δn−2).

Therefore, the polynomial has the same number of roots in (0,1) as P(δ)=1−2δ+δn−1, which by Descartes’ Rule of Signs has either zero or two positive roots. One of them is obsviously for δ=1 and and the other is for some δ^<1, because the polynomial has a unique minimum for some δ<1 and is positive for δ=0. Moreover, note that P(1/2)>0 for any n, and in fact P(1/2)→0 as n→∞. Hence, the expression is always positive in (0,1/2], which turn means that A1+A2δ+⋯+An−1δn−2 has no root for δ∈(0,1/2] and in fact it is always positive in that region, by continuity.

4. The proof is identical to that of condition 3, if one considers the alternative expression of difference between decay centralities, DCi(δ)−DCj(δ), that is provided by eq. (2).

Proof of Proposition 8

1. Consider again the reformulated expression with ϵ=1−δ, for which we know that DCi(ϵ)−DCj(ϵ)=−ϵ[B1+B2ϵ+⋯+Bn−1ϵn−2], for which it is sufficient to show that it has no root for ϵ∈(0,1/2] as long as |B1|≥|Bl| for all l∈{2,…,n−1}. This is proven identically to condition 3 of Proposition 7. The obtained result, together with the fact that B1<0 ensures that DCi(ϵ)>DCj(ϵ) for all ϵ∈(0,1/2], which is identical to saying that DC(δ)>DCj(δ) for all δ∈[1/2,1).

2. The proof is identical to that of condition 1, if one considers the alternative expression of difference between decay centralities, DCi(ϵ)−DCj(ϵ), that is provided by eq. (3).