SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Example 1. Consider a random network generated from the stochastic block model with node partitions  $[N_A, N_B, N_C, N_D] = [100, 100, 10, 10]$  and block matrix shown in Figure 1 left. The diagram in Figure 1 middle illustrates the network: A, B are the two main clusters with high link density (0.5); the other two clusters, C, D, are much smaller and have lower link densities (0.1, 0.3) both internally and to A, B. Structurally speaking, C, D serve as two "bridges" between A and B.
Now consider adding links to three groups of previously disconnected node pairs defined as:
- Group 1: both end nodes in Cluster A, e.g.  $(a_{3}, a_{4})$
- Group 2: one end node in Cluster A, one end node in cluster B, both linked to some other node(s) in cluster C, e.g.  $(a_{1}, b_{1})$
- Group 3: one end node in Cluster A, one end node in cluster B, both linked to some other node(s) in cluster D, e.g.  $(a_{2}, b_{2})$
Our simulation shows that Group 1 introduces least conflict reduction on average — in fact much less than the other two groups do. Since each pair of nodes in Group 1 are from the same densely connected cluster, it means that the numerator term  $\sum_{k\in V}(\mathcal{N}_{ik} - \mathcal{N}_{jk})^2$  dominates  $\Delta_{+e}\mathbb{E}_s[\mathcal{C}]$ . Therefore, we may conclude that in general a link reduces more conflict if it involves two nodes that are globally distant. Comparing Group 2 and 3, we can further see that node pairs in Group 2 have stronger local connectivity than those in Group 3 ("bridge" C has higher link densities than "bridge" D). The fact that Group 2 reduces more conflict on average shows that stronger local connectivity actually positively contribute to conflict minimization.
We have now found the following two features of conflict-minimizing links, when initial opinions  $s$  are sampled from a distribution with finite variance:
1. Globally, both end nodes should belong to different clusters, so that  $\sigma^2 |(I + L)^{-1}b_e|_2^2$  is large;
2. Locally, both end nodes should be decently well-connected with each other, so that  $1 + b_{e}^{T}(I + L)^{-1}b_{e}$  is small.
4.2 Relating Conflict Minimization to Relevance
Relevant links are links that are likely to be accepted by users. Over the past two decades, relevant links have been extensively studied as the core subject of link prediction problem in many different contexts [11-13]. We know that the relevance of social links has strong correlation with small graph distance between the two end nodes [12,40-45]. Comparing this existing knowledge with our
characterizations of conflict-minimizing links, it is not hard to perceive that relevance and conflict minimization are not always incompatible with each other. Instead, they can have a decent degree of alignment in some cases.
5 Measuring the Degree of Alignment Between Relevance and Conflict Minimization on Real-World Data
Sec. 4's analysis shows that the two features of relevance and conflict minimization are not strictly incompatible with each other. This section discusses how we can measure the two features' degree of alignment on real-world data.
5.1 Definition of Conflict Awareness
We start by formulating link additions: A link addition function  $f$  is defined as:  $f(e;G,\beta):(V\times V)\to [0, + \infty)$ , where the function parameters are a given social network  $G = (V,E)$  and a budget  $\beta$  for adding links.  $f$  is defined on node pairs, subject to the budget constraint  $\sum_{e\in V\times V}f(e;G,\beta) = \beta$ .
Among all possible link addition functions, there is a conflict-minimizing function  $f^{*}(e;G,\beta)$ , which is the function that reduces the most conflict under budget  $\beta$ . We use  $\Delta_f\mathcal{C}$  and  $\Delta_f\mathbb{E}_s[\mathcal{C}]$  to denote the conflict change and expected conflict change caused by applying  $f$  over the network  $G$ . The two terms are related by  $\Delta_f\mathbb{E}_s[\mathcal{C}] \equiv \int_s\rho (s)\Delta_f\mathcal{C}d_s$ , where  $\rho (s)$  is the probability density function of  $s$ . We further use  $L_{f}$  to denote the Laplacian of the network formed by only the new links added by  $f$ .
Definition 1. Given a social network  $G$ , initial opinions  $s$ , and a positivie budget  $\beta$ , the conflict awareness (CA) of a link addition function  $f(e;G,\beta)$  is defined by the conflict reduced by applying  $f$  to add links, divided by the best possible conflict reduction by applying  $f^{*}$  to add links:
$$
\boldsymbol {C} \boldsymbol {A} (f) = \Delta_ {f} \mathcal {C} / \Delta_ {f ^ {*}} \mathcal {C} \tag {8}
$$
$$
\Delta_ {f} \mathcal {C} = s ^ {T} (I + L + L _ {f}) ^ {- 1} s - s ^ {T} (I + L) ^ {- 1} s \tag {9}
$$
$$
\Delta_ {f ^ {*}} \mathcal {C} = \min  _ {L _ {f}} \quad \Delta_ {f} \mathcal {C} \tag {10}
$$
$$
\text {s u b j e c t} L _ {f} \in \mathcal {L} (\text {L a p l a c i a n c o n s t r a i n t}) \tag {11}
$$
$$
\operatorname {T r} \left(L _ {f}\right) \leq 2 \beta (\text {b u d g e t c o n s t r a i n t}) \tag {12}
$$
$L + L_{f}$  is the Laplacian matrix of the network after being modified by  $f$ , so the definition of  $\Delta_f\mathcal{C}$  is straightforward.  $\Delta_{f^*}\mathcal{C}$  is defined as the objective of an optimization problem, which essentially looks for the best network with total link weights  $\beta$  to be superimposed over the original network  $G$ .
The measure conflict awareness is useful for two reasons. First, CA essentially measures how "bad" any link recommendation algorithm is with regards to minimizing opinion conflict. By assigning  $f$  a function that recommends relevant links, CA immediately becomes a quantifier that shows how much mismatch exists between  $f$ -suggested relevant links and conflict-minimizing links. Second, CA is a better measure than the pure conflict change  $\Delta_f\mathcal{C}$ . This is because CA is normalized to [0,1], which allows meaningful comparisons across different social networks. We will see that this property becomes especially helpful in practice for characterizing a link recommendation algorithm.
We can further show that CA also has a computationally desirable feature of being convex.
Proposition 1. In Definition 1,  $\Delta_{f^*}\mathcal{C}$  is the objective of a convex optimization problem.
We can generalize Definition 1 to expectation of conflict awareness over a distribution of initial opinions  $s$  and prove its convexity. See Appendix B.6 for details.
5.2 Measuring Conflict Awareness on Real-World Social Networks
5.2.1 Motivations and Experimental Setup
We use the measure of conflict awareness defined in Sec. 5.1 to empirically investigate two important questions on the relationship between relevance and conflict reduction in link recommendations:
- What is the conflict awareness of some of the popular off-the-shelf link recommendation algorithms? High conflict awareness (e.g. close to 1.0) means that the algorithm is effective at reducing conflict, and low conflict awareness (e.g. less than 0.2) means the opposite.
Figure 2: Measurement of conflict awareness and recall for 13 link recommendation algorithms on samples of Reddit and Twitter social networks. The x-axis is the  $\eta = \frac{\#negative\ links}{\#positive\ links}$  that controls class imbalance in the test set.
- Does a more accurate link recommendation algorithm have higher conflict awareness? If we observe that for most recommendation algorithms the answer is affirmative, it means that relevance and conflict reduction are positively correlated; if we observe the opposite, it means relevance and conflict reduction are incompatible in practice.
Datasets. We use two real-world datasets: Reddit and Twitter, collected by [39], one of the pioneering works that conduct empirical studies on the FJ model. See Appendix C.3 for more details on how the network data and the initial opinions are generated for the two datasets.
Baselines. We test three groups of 13 different link recommendation methods.
- Unsupervised distance-based measures: Personalized PageRank [46], Katz Index [47], Jaccard Index [48], Adamic Adar [49], Common Neighbors [50], Preferential Attachment [50], Resource Allocation [51]. They are among the most classic methods for link recommendations, whose distance-based heuristics underpin many deep-learning-based link recommendation methods.
- Self-supervised graph learning: Logistic Regression (with Node2Vec [52] embeddings as input node features), Graph Convolutional Neural Network (GCN) [53], Relational Graph Convolutional Nerual Network (R-GCN) [54], Graph Transformer [55], SuperGAT [56]. The last three one are all GNN-based methods that achieved previous state-of-the-art on link prediction task.
- Conflict minimization solver: This is to solve the convex optimization problem defined by Eq.(10)-(12); they find optimal weights for links to be added with the sole goal of minimizing the conflict. However, it is crucial to note that this solver has total ignorance of relevance, i.e. it can't differentiate positive and negative links apart. Therefore, it may actually end up recommending many negative (i.e. invalid) links that have no effect on conflict.
Evaluation pipeline. For each dataset, we randomly sample  $\beta = 100$  positive links from the edge set, and  $\beta \cdot \eta$  negative links from all disconnected node pairs; the negative sampling rate  $\eta \in [1,10]$  is a hyperparameter. The positive links are then removed from the network to be reserved for testing, together with the negative links. It is crucial to note that in this setting (and in the real world), only a positive link can be added to the social network. If a negative link gets recommended (i.e. assigned positive weights), it can't be added to the network. The whole process is repeated for 10 times with different random seeds.
All links in the original network have unit weights, although the recommender is allowed to assign continuous weights to new links. As there are  $\beta$  positive links, we require that the total recommended weights must sum up to  $\beta$ , which is enforced through linear scaling of  $f$ 's output. See Appendix C.5.
Evaluation metrics. Besides conflict awareness, we also measure the recall and precision@10 of each link recommendation method as proxies for "relevance".
Reproducibility: Our code and data can be downloaded from here. Other configurations of the numerical experiment can be found in Appendix C.4
5.2.2 Result Analysis
Fig.2 shows the measurement results of both conflict awareness and recall rate for the 13 link recommendation algorithms on the two datasets. The precision@10 measurement results are in Appendix C. The y-axis is the negative sampling rate  $\eta$ . The results allow us to make the following observations.
First, we observe that the conflict awareness can vary a lot for different algorithms under different settings. For example, on Reddit's social network, R-GCN's conflict awareness can be as high as 0.95 with  $\eta = 1$ , while Jaccard Index's conflict awareness is mostly below 0.2.
Second, we examine the effect of  $\eta$ . In principle, a larger  $\eta$  means it is harder to identify positive links in the test set. For both networks, we can see that the conflict awareness of most algorithms drop as  $\eta$  goes up, though the four GNN-based methods seem to be slightly less affected. Similar trends can be observed from the recall plots (b) and (d), where GNN-based algorithms stay quite robust as the task gets increasingly harder. These observations suggest that the ability to suggest "relevant" links can be crucial for maintaining good conflict awareness, especially when the recommendation task is hard.
Third, we have interesting observations on the "Conflict Minimization" algorithm: on both networks, this algorithm consistently produces the worst recall among all algorithms — barely usable from the perspective of relevance. However, although there are not many relevant links in its recommendation, the algorithm still has the best conflict awareness most of the time. among the algorithm's recommended links, the few ones that are actually relevant (positive) are highly effective to reduce conflict. In that sense, we can still clearly see the there exists a certain amount of misalignment between "relevance" and "conflict reduction".
6 Limitation: the Paradox of Conflict and Happiness
We found a limitation in the long line of existing works analyzing opinion conflict based on the FJ model, which our work also inherits from. The conflict measure we've discussed so far may not reflect user's happiness in their social engagement. Originally proposed in the seminal work [57], the happiness measure (or equivalently the unhappiness) quantifies the amount of mental pressure felt by people in their social engagement, as the sum of two terms: the amount of disagreement with friends (i.e. our disagreement term), and the amount of opinion shift between initial opinions and expressed opinions, i.e. internal conflict, defined as
$$
\mathcal {I} (G, s) = \sum_ {i \in V} \left(z _ {i} - s _ {i}\right) ^ {2} = s ^ {T} \left(\left(I + L\right) ^ {- 1} - I\right) ^ {2} s
$$
We use  $\mathcal{U}(G,s)$  to denote the unhappiness of people over a social network  $G$  and initial opinions  $s$ . Fig. 3 illustrates the relationship of important concepts in the FJ model we've discussed so far.
Fig.3 shows that the common term, disagreement, is shared by both measures of polarization and disagreement. Therefore, it seems likely that when one measure changes (say conflict drops due to the addition of a new link), the other measure should change in the same direction. Surprisingly however, this is not true. In fact, the following theorem shows the two measures to always change in opposite directions when network structure changes!
Figure 3: Relationship of the important concepts in the Friedkin-Johnsen opinion model.
Theorem 5. Given initial opinions  $s$  and social network  $G = (V, E)$ , we have the following conservation law about conflict and unhappiness (notice the RHS is independent of the graph structure):
$$
\mathcal {C} (G, s) + \mathcal {U} (G, s) = s ^ {T} s \tag {13}
$$
Theorem 5 reveals a paradox: reducing conflict by modifying the structure of a social network always comes at the expense of more unhappiness of people. To resolve this paradox goes beyond this paper's scope, but would be a very interesting topic to study in the future.
In this work, we analyzed the relationship between relevance and opinion conflict in online social link recommendations. We present multiple pieces of evidence challenging the view that the two objectives are totally incompatible in link recommendations. For future work, it would be extremely interesting to study recommendation algorithms that can combine the two features. We also conjecture that rigorous bounds can be derived for the conflict awareness of some classical link recommendation methods such as Peronalized PageRank and Katz Index.
The authors thank Eva Tardos and Sigal Oren for their helpful feedback on this work. This work is supported in part by a Simons Investigator Award, a Vannevar Bush Faculty Fellowship, MURI grant W911NF-19-0217, AFOSR grant FA9550-19-1-0183, ARO grant W911NF19-1-0057, a Simons Collaboration grant, and a grant from the MacArthur Foundation.
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Our code and data can be downloaded from https://github.com/Abel0828/NeurIPS23-Conflict-Relevance-in-FJ.
To mitigate this risk, we suggest social platforms take more cautious steps when deciding to reduce the exposure of one person's content feed to another, such as additional algorithmic check in background, as well as more security measures to guard against the hacking of platform's administrative authority. Researchers are also encouraged to study network structures that are more robust to attacks of such kind, as well as defense measures to be taken when such attacks actually happen.
Proof. Let  $L_{+e}$  denote the Laplacian matrix of the new social network after adding a new link  $e = (i,j)$ . To prove Eq.(2), we invoke the Sherman-Morrison Formula [58] for computing the inverse of rank-1 update to an invertible matrix. Notice that  $G_{+e} = L + b_{e}b_{e}^{T}$ . Therefore,
$$
\begin{array}{l} \mathcal {C} (G _ {+ e}, s) - \mathcal {C} (G, s) = s ^ {T} \left(\left(I + G _ {+ e}\right) ^ {- 1} - \left(I + L\right) ^ {- 1}\right) s \\ = s ^ {T} \left(\left(I + L + b _ {e} b _ {e} ^ {T}\right) ^ {- 1} - \left(I + L\right) ^ {- 1}\right) s \\ = - s ^ {T} \frac {\left(I + L\right) ^ {- 1} b _ {e} b _ {e} ^ {T} \left(I + L\right) ^ {- 1}}{1 + b _ {e} ^ {T} \left(I + L\right) ^ {- 1} b _ {e}} s \\ = - \frac {\left| b _ {e} ^ {T} (I + L) ^ {- 1} s \right| _ {2} ^ {2}}{1 + b _ {e} ^ {T} (I + L) ^ {- 1} b _ {e}} \\ = - \frac {\left(z _ {i} - z _ {j}\right) ^ {2}}{1 + b _ {e} ^ {T} (I + L) ^ {- 1} b _ {e}}. \\ \end{array}
$$
$L$  is positive semidefinite, so  $(I + L)^{-1}$  is also positive semidefinite. Therefore,  $1 + b_{e}^{T}(I + L)^{-1}b_{e}$  is positive, and so  $-\frac{(z_i - z_j)^2}{1 + b_e^T(I + L)^{-1}b_e}\leq 0.$  To prove Eq.(3), we further note that
$$
\begin{array}{l} \mathbb {E} _ {s} \left[ \mathcal {C} (G _ {+ e}, s) - \mathcal {C} (G, s) \right] = \mathbb {E} _ {s} \left[ - s ^ {T} \frac {(I + L) ^ {- 1} b _ {e} b _ {e} ^ {T} (I + L) ^ {- 1}}{1 + b _ {e} ^ {T} (I + L) ^ {- 1} b _ {e}} s \right] \\ = \mathbb {E} _ {s} \left[ - \frac {b _ {e} ^ {T} (I + L) ^ {- 1} s s ^ {T} (I + L) ^ {- 1} b _ {e}}{1 + b _ {e} ^ {T} (I + L) ^ {- 1} b _ {e}} \right] \\ = - \frac {b _ {e} ^ {T} (I + L) ^ {- 1} \mathbb {E} _ {s} [ s s ^ {T} ] (I + L) ^ {- 1} b _ {e}}{1 + b _ {e} ^ {T} (I + L) ^ {- 1} b _ {e}} \\ = - \frac {b _ {e} ^ {T} (I + L) ^ {- 1} (\sigma^ {2} I) (I + L) ^ {- 1} b _ {e}}{1 + b _ {e} ^ {T} (I + L) ^ {- 1} b _ {e}} \\ = - \frac {\sigma^ {2} | (I + L) ^ {- 1} b _ {e} | _ {2} ^ {2}}{1 + b _ {e} ^ {T} (I + L) ^ {- 1} b _ {e}} \leq 0. \\ \end{array}
$$
Proof. Let  $M = I + L$ , and let matrix  $C$  be the co-factor matrix of  $M$ , then  $(I + L)^{-1} = M^{-1} = |M|^{-1}C$ . Therefore,  $b_{e}^{T}(I + L)^{-1}b_{e} = |M|^{-1}\big(C_{ii} + C_{jj} - C_{ij} - C_{ji}\big)$ .  $|M|$  is the determinant of matrix  $M$ . [59] presents a result that  $|M|$  equals the total number of spanning rooted forests of  $G$ , and  $C_{xy}$  equals the total number of spanning rooted forests of  $G$ , in which node  $x$  and  $y$  belong to the same tree rooted at  $x$ . The theorem is proved by substituting this previous result back into  $|M|^{-1}\big(C_{ii} + C_{jj} - C_{ij} - C_{ji}\big)$ .
$$
\begin{array}{l} \sigma^ {2} | (I + L) ^ {- 1} b _ {e} | _ {2} ^ {2} = \sigma^ {2} \sum_ {k \in V} \left(| M | ^ {- 1} C _ {i k} - | M | ^ {- 1} C _ {j k}\right) ^ {2} \\ = \sigma^ {2} | M | ^ {- 2} \sum_ {k \in V} \left(C _ {i k} - C _ {j k}\right) ^ {2} \\ \end{array}
$$
Since  $M$  is symmetric, we have  $C_{ik} + N_{ik} = C_{ki} + \mathcal{N}_{ki} = C_{kk}$ ,  $C_{jk} + \mathcal{N}_{jk} = C_{kj} + \mathcal{N}_{kj} = C_{kk}$ , where  $C_{kk}$  according to [59] is equal to the total number of spanning rooted forests where node  $k$  is at the root of the tree to which  $k$  belongs. Joining the two equations, we have  $C_{ik} - C_{jk} = \mathcal{N}_{ik} - \mathcal{N}_{jk}$ . Therefore,
$$
\sigma^ {2} | (I + L) ^ {- 1} b _ {e} | _ {2} ^ {2} = \sigma^ {2} \mathcal {N} ^ {- 2} \sum_ {k \in V} (\mathcal {N} _ {i k} - \mathcal {N} _ {j k}) ^ {2}
$$
B.4 Proof of Corollary 1
Proof. The correctness quickly follows from substituting Equations (5, 6) into Equation (2).
B.5 Proof of Proposition 1
Proof. To show that the objective is convex, we resort to the result in [60], Example 9:  $X^{-1}$  is a matrix convex on the set of all nonnegative invertible Hermitian matrices. Obviously  $I + L + L_{f}$  is nonnegative, invertible and symmetric, so it is a matrix convex. Therefore, the objective is convex. Any convex combination of Laplacians is still a Laplacian. The trace of any convex combination of of matrices cannot exceed the trace of any members. Therefore, the feasible region is also convex.
B.6 Expected Conflict Awareness
Definition 2. Given a social network  $G$  and a budget  $\beta > 0$ , the conflict awareness over Expectation (CAE) of a link addition function  $f(e;G,\beta)$  is likewise defined as:
$$
\mathbf {C A E} (f) \equiv \frac {\Delta_ {f} \mathbb {E} _ {s} [ \mathcal {C} ]}{\Delta_ {f ^ {*}} \mathbb {E} _ {s} [ \mathcal {C} ]} \tag {14}
$$
$$
\Delta_ {f} \mathbb {E} _ {s} [ \mathcal {C} ] \equiv \sigma^ {2} \left[ \operatorname {T r} \left(\left(I + L + L _ {f}\right) ^ {- 1}\right) - \operatorname {T r} \left(\left(I + L\right) ^ {- 1}\right) \right] \tag {15}
$$
$$
\Delta_ {f ^ {*}} \mathbb {E} _ {s} [ \mathcal {C} ] \equiv \min  _ {L _ {f}} \quad \Delta_ {f} \mathbb {E} _ {s} [ \mathcal {C} ] \tag {16}
$$
subject to  $L_{f}\in \mathcal{L}$  (Laplacian constraint) (17)
$$
\operatorname {T r} \left(L _ {f}\right) \leq 2 \beta (\text {b u d g e t c o n s t r a i n t}) \tag {18}
$$
Proposition 2. The definition of  $\Delta_f\mathbb{E}_s[\mathcal{C}]$  above is consistent with that of  $\Delta_f\mathcal{C}$  in Definition 1 in the sense that they satisfy  $\Delta_f\mathbb{E}_s[\mathcal{C}]\equiv \int_s\rho (s)\Delta_f\mathcal{C}d_s$
Proof. Let  $A$  be any square matrix of the same shape as  $L$ . Then  $\int_{s} \rho(s) s^{T} A s d_{s} = \int_{s} \rho(s) s^{T} (A s) d_{s} = \int_{s} \rho(s) \mathrm{Tr}((A s) s^{T}) d_{s} = \int_{s} \rho(s) \mathrm{Tr}(A (s s^{T})) d_{s} = \mathrm{Tr}(A \int_{s} \rho(s) (s s^{T}) d_{s}) = \mathrm{Tr}(A (\sigma^{2} I)) = \sigma^{2} \mathrm{Tr}(A)$ . By substituting  $A = (I + L + L_{f})^{-1}$  and  $A = (I + L)^{-1}$  into Eq. (9) respectively, the proposition is proved.
Proposition 3. In Definition 2,  $\Delta_{f^*}\mathbb{E}_s[\mathcal{C}]$  is also the objective of a convex optimization problem.
Proof. From the proof for Proposition 1, it suffices to only show that the  $\Delta_f\mathbb{E}_s[\mathcal{C}]$  in Equation 15 is convex in  $L_{f}$  given other variables fixed. Notice that we mentioned  $\Delta_f\mathbb{E}_s[\mathcal{C}]\equiv \int_s\rho (s)\Delta_f\mathcal{C}d_s,$  in which  $\rho (s)\geq 0$ $\Delta_f\mathcal{C}$  can be viewed as a function of  $L_{f}$  and  $s$  , and is convex in  $L_{f}$  given  $s$  to be further fixed. Therefore, the integral  $\Delta_f\mathbb{E}_s[\mathcal{C}]$  is also convex in  $L_{f}$
Proof. Let  $0 = \lambda_1 \leq \lambda_2 \leq \ldots \leq \lambda_n$  be eigenvalues of  $L$  in ascending order; the eigen decomposition of  $L = U\Lambda U^T$  where  $\Lambda = \mathrm{diag}\big([ \lambda_1, \dots, \lambda_n ] \big)$  and  $U$  is the corresponding orthonormal matrix satisfying  $U U^T = I$ . Notice that  $(I + L)^{-1} = U(I + \Lambda)^{-1}U^T$ .
$$
\frac {\mathcal {C} (G _ {0} , s)}{\mathcal {C} (G , s)} = \frac {s ^ {T} s}{s ^ {T} (I + L) ^ {- 1} s} = \frac {s ^ {T} U U ^ {T} s}{s ^ {T} U (I + \Lambda) ^ {- 1} U ^ {T} s}
$$
let  $s' = U^T s$ , and further notice that since we have assumed  $s$  to be zero-centered (see Section2),  $s_1' = 1^T s = 0$ . We can further rewrite:
$$
\frac {\mathcal {C} (G _ {0} , s)}{\mathcal {C} (G , s)} = \frac {s ^ {\prime T} s ^ {\prime}}{s ^ {\prime T} (I + \Lambda) ^ {- 1} s ^ {\prime}} = \frac {\sum_ {i = 1} ^ {n} s _ {i} ^ {\prime 2}}{\sum_ {i = 1} ^ {n} (1 + \lambda_ {i}) ^ {- 1} s _ {i} ^ {\prime 2}} = \frac {\sum_ {i = 2} ^ {n} s _ {i} ^ {\prime 2}}{\sum_ {i = 2} ^ {n} (1 + \lambda_ {i}) ^ {- 1} s _ {i} ^ {\prime 2}}
$$
It is not hard to see that
$$
1 + \lambda_ {n} \geq \frac {\mathcal {C} (G _ {0} , s)}{\mathcal {C} (G , s)} \geq 1 + \lambda_ {2}
$$