Minimizing Hitting Time between Disparate Groups with Shortcut Edges

Structural bias or segregation of networks refers to situations where two or more disparate groups are present in the network, so that the groups are highly connected internally, but loosely connected to each other. In many cases it is of interest to increase the connectivity of disparate groups so as to, e.g., minimize social friction, or expose individuals to diverse viewpoints. A commonly-used mechanism for increasing the network connectivity is to add edge shortcuts between pairs of nodes. In many applications of interest, edge shortcuts typically translate to recommendations, e.g., what video to watch, or what news article to read next. The problem of reducing structural bias or segregation via edge shortcuts has recently been studied in the literature, and random walks have been an essential tool for modeling navigation and connectivity in the underlying networks. Existing methods, however, either do not offer approximation guarantees, or engineer the objective so that it satisfies certain desirable properties that simplify the optimization~task. In this paper we address the problem of adding a given number of shortcut edges in the network so as to directly minimize the average hitting time and the maximum hitting time between two disparate groups. Our algorithm for minimizing average hitting time is a greedy bicriteria that relies on supermodularity. In contrast, maximum hitting time is not supermodular. Despite, we develop an approximation algorithm for that objective as well, by leveraging connections with average hitting time and the asymmetric k-center problem.


INTRODUCTION
The last decade has seen a surge in the development of methods for detecting, quantifying, and mitigating polarization and controversy in social media.An example of a polarized network is a conversation or endorsement graph associated with a controversial topic on Twitter.The nodes represent users, and the edges interactions between users in the form of posts, likes, endorsements, etc., related to a controversial topic.It has been observed that the interactions typically occur between like-minded individuals, resulting in the reinforcement of ones own beliefs [7,12].
Structural bias arises in many types of networks beyond social networks, such as content and information networks.Recently, several methods have been developed for reducing structural bias [24] or segregation [13,15] in content networks.An example of a content network is the network obtained by "what to watch next" recommendations in a video-sharing platform.Content in such networks can often be divided into two or more groups, each of which is highly connected internally, but loosely connected to each other.These groups could be the result of differentiating between "harmful" (or radicalized) content and "neutral" (or non-radicalized) content, or simply the result of different opinion (pro/contra) on a certain issue.It might be beneficial for users navigating these platforms to be exposed to diverse content-exposing themselves to multiple viewpoints-in order to become better informed.
A particular line of research focuses on adding new edges (denoted as shortcut edges) to a network, with the goal of reducing some quantifiable measure of polarization, controversy, structural bias, or segregation [13-15, 21, 24, 45].Many of these measures are defined on the basis of random walks between groups of nodes.In content graphs, random walks are a natural and simple model for how a user navigates through content.For example, the Random Walk Controversy (RWC) score of Garimella et al. [21] has been reported to "discriminate controversial topics with great accuracy" in Twitter conversation graphs.The authors in that work suggest a method that adds a fixed number of shortcut edges to a graph in order to reduce the RWC score of the augmented graph [21].
However, most of these random walk-based augmentation problems do not have algorithms with provable approximation guarantees [21], or the optimized measure has been reformulated so that it exhibits a desirable property [13,15,24].
For example, the work of Haddadan et al. [24] considers maximizing the gain function Δ  ( ) = (∅) − ( ) in the context of bounded-length random walks in directed graphs.Here,  is a set of at most  new shortcut edges that will be added to an input graph, and the function  is the average expected hitting time for a random walk starting in one group to hit the other group (see Definition 1).They prove that Δ  is a non-negative monotone submodular set function, and thus, the well-known greedy algorithm of [43] ensures a (1 − 1/) approximation guarantee.
Similarly, the work of Fabbri et al. [15] considers a gain function in the form of Δ  (•) =  (∅) −  (•) in the context of degree-regular directed graphs and with rewiring operations instead of shortcut edge additions.Here, the function  is the largest hitting time instead of the average (see Definition 1).Fabbri et al. show that the existence of a multiplicative approximation algorithm implies P = NP.This result leads them to propose heuristic algorithms.However, their inapproximability result does not follow from the use of random walks, nor from the rewire operations.Their reduction still applies if one would replace "largest hitting time" with "largest shortest path distance, " for example.Hence, their inapproximability result is an artefact of maximizing a gain function in combination with the max operator in the function  (Definition 1).
The previous discussion motivates the following question: "What can we say about directly minimizing the functions  and , instead of indirectly optimizing them by maximizing associated gain functions?" This paper provides several first algorithmic results and ideas regarding this question for both functions  and .We take an abstract view and define our problems on input graphs in their simplest form, without the additional constraints (bounded-length random walks or degree-regular graphs) considered in Haddadan et al. [24] and Fabbri et al. [15], respectively.We consider uniform simple random walks of unbounded length on undirected, unweighted and connected input graphs.

Results and techniques.
For the first problem we study, minimize average hitting time (BMAH), we observe that the objective is supermodular and it can be optimized by a greedy bicriteria strategy [32], which offers an (1+)-approximation at the cost of adding logarithmically more edges.In addition, we show how to speed up this algorithm by deriving an approximation for the objective, which relies on sampling a small number of bounded-length absorbing random walks and bounding the error term using eigenvalue techniques.The length of these walks directly depends on the mixing time and average degree of the red nodes, which are bounded quantities in most real-life networks.We then show that running the same greedy strategy of Liberty and Sviridenko [32] on the approximative values, still provides a (2 + )-approximation factor.
For our second problem, minimize maximum hitting time (BMMH), the objective function is neither supermodular nor submodular, and thus, it is a more challenging task.Nevertheless, we present two different algorithms, both with approximation guarantees, albeit, weaker than in the case of average hitting time.The first algorithm utilizes a relation between maximum and average hitting time, and uses the greedy strategy designed for the BMAH problem.The second algorithm leverages a novel connection between BMMH and the asymmetric -center problem [26], and utilizes optimal methods developed for the latter problem [4].

RELATED WORK
A large body of work has been devoted to designing methods for optimizing certain graph properties via edge additions, edge rewirings, or other graph-edit operations.These approaches include methods for increasing graph robustness [10], maximizing the centrality of a group of nodes [37], improving the betweenness centrality of a node [8], reducing the average shortest path distances over all pairs of nodes [39,45], minimizing the diameter of the graph [1,14], increasing resilience to adversarial attacks [35], and more.Some of these approaches provide algorithms with provable approximation guarantees, while the others are mostly practical heuristics.
The setting of the problem we consider has also been studied in the context of understanding phenomena of bias, polarization, and segregation in social networks [6,11,12,17,21,23,40,48], and developing computational methods to mitigate those adverse effects [2,19,20,22,36,42,[50][51][52].Most of the works listed above seek to optimize a complex objective related to some dynamical diffusion process, such as information cascades [28], or opinion dynamics [18].In all cases, the optimization functions studied in previous work are significantly different from the hitting-time objectives considered in this paper.
As discussed in the introduction, the work that is most closely related to ours, is the paper by Haddadan et al. [24], which seeks to reduce the structural bias between two polarized groups of a network via edge insertions.Their approach is based on hitting time for bounded-length random walks, but contrary to our proposal, they seek to maximize the reduction of hitting time caused by edge insertions, rather than directly minimizing the hitting time.Similar ideas have been proposed by other authors, e.g., Fabbri et al. [15] and more recenty Coupette et al. [13], in the context of reducing exposure to harmful content.We consider our work to be a compelling extension of such previous ideas, not only due to optimizing a more natural objective, but also for coping with a more challenging optimization problem, for which our solution offers new technical ideas and insights.
[ ] be the subset of edges of  that have both endpoints in  , and  [ ] = (,  [ ]) be the subgraph of  induced by  .If  is a set of non-edges of  = ( , ), then  +  denotes the graph ( ,  ∪  ).The degree of a node  in , which is the number of neighbors of  in , is denoted as   ().We let   be the average of the degrees of red nodes.Random walks.Every random walk will be a uniform simple random walk, unless stated otherwise.By uniform simple random walk we mean that at each step one of the neighbors of the current node is selected with uniform probability.Given a graph  = ( , ) and  ⊆  , the random variable   () indicates the first time that a random walk on  starting from  visits .If  ∈ , we set   () = 0.The expectation of this random variable is denoted as where the subscript emphasizes that the random walk is on .With a slight abuse of notation,   (, ) is thus the (expected) hitting time for a walk starting from  to visit  for the first time.Note that   (, ) = 0 and   (, ) = 0 if and only if  ∈ .It also holds that   (, ) ≤   (, ) when  ⊆ , and in particular   (, ) ≤ min  ∈   (, ).Matrices and norms.We denote A as the adjacency matrix of a graph, and D is the degree diagonal matrix.Let   ∈ R  ×1 be the vector of all-ones of size  , and let ∥•∥ 2 denote the spectral norm.Additional definitions.For a graph  = ( , ) with a valid bipartition  = {, }, let ( × ) \  be the set of inter-group non-edges in .Define the following two set functions  ,  : 2 (×)\ → R >0 as the maximum and average hitting time.

PROBLEMS AND OBSERVATIONS
In this section, we formally define the problems we study, and we discuss their properties.

Problems
We introduce and study the following two problems, which we denote as the BMMH (Budgeted Minimum Maximum Hitting time) and BMAH (Budgeted Minimum Average Hitting time) problems.
Problem 1 (Budgeted Minimum Maximum Hitting time problem (BMMH)).Given an undirected connected graph  = ( , ), with a valid bipartition  = {, }, and a budget  ∈ N, we seek to find a set of inter-group non-edges  ⊆ ( × ) \  with | | ≤  that minimizes  ( ), where  is the maximum hitting-time function defined in Definition 1.
Problem 2 (Budgeted Minimum Average Hitting time problem (BMAH)).Identical definition to Problem 1, but we aim to minimize the function , instead of  , where  is the average hitting-time function defined in Definition 1.
BMMH aims to augment  with  new inter-group edges, so as to minimize the largest hitting time to a blue node, for random walks starting from red nodes.Likewise, BMAH aims to minimize the average hitting time from red to blue nodes.Note that for every red node we allow multiple new edges to be incident to it, as long as the augmented graph remains simple.

Observations
The NP-hardness of both BMMH and BMAH follows from a reduction from the minimum set cover problem, very similar to the reduction detailed by Haddadan et al. [24] and omitted for brevity.We present several useful observations regarding BMMH and BMAH that will guide the design of approximation algorithms: Observation 1. Changing the blue endpoints of the edges in a feasible solution  does not change  ( ) or ( ).
Proof.Every walk starts from a red node and halts the moment a blue node is hit, hence the identity of the blue endpoints of edges in  do not matter.□ Both problems thus reduce to selecting a multiset of red endpoints, as we allow multiple edges in  to have the same red endpoint.If a red node is connected to all the blue nodes already, we can of course not add any more new edges incident to it.
Proof.Let  be a feasible solution.For every  ∈ , the hitting time  + (, ) is a monotonically decreasing set function of  , since the edges in  are inter-group edges.Every  ∈  only helps in reducing the hitting time from  to .This can be proven more formally by a straightforward coupling argument between a walker on  and  + [24,34].Since individual hitting times are monotone, the functions  and  are monotone, as well.□ The following supermodularity property for individual hitting times is less obvious, and a variant of it was proven by Haddadan et al. [24] for bounded-length random walks on directed graphs.We give a different and shorter proof of supermodularity in our case of unbounded random walks, by using a coupling argument.
Observation 3 (supermodularity of hitting times).For all  ∈  and for all Proof.Pick any  ∈ .We assume all walks start from  .Consider the right-hand side of Eq. ( 1).Couple walks on  +  1 to walks on  by simply following the walker on , until the walker hits the red endpoint  1 ∈  of the edge  1 .If the walker hits  1 , follow the edge  1 with probability 1   ( 1 )+1 .With probability ( 1 )+1 , keep following the walker on .It is clear that the marginal distribution of this coupled walk is a uniform random walk on  +  1 .
As long as  1 is not traversed, the two coupled walks are identical.If  1 is traversed, the walks are identical up until the time prior to traversing  1 , when they are both in  1 for the last time.On this moment, the walker on  still needs to travel   ( 1 , ) steps on Algorithm 1 [Greedy] bicrit.(1 + )-approximation for BMAH.
average to hit , while the walker on  +  1 needs 1 more step (conditioned on  1 being traversed).Hence for all  ∈  we have Similarly, we can write for the left-hand side of Eq. ( 1)

□
Observation 3 states that individual hitting times are supermodular, hence the objective function  of BMAH is also supermodular.However, the objective function  of BMMH is not guaranteed to be supermodular, as a result of the max operator.Observation 4 shows that even the weaker notion of weakly- supermodularity [33] does not hold for  .
Observation 4. The function  is supermodular, but the function  is not weakly-supermodular for any  ≥ 1.
Proof.The supermodularity of  is direct, as it is an average of supermodular functions (Observation 3).We prove the second statement of the observation.A non-negative non-increasing set function  on [] is said to be weakly- supermodular with  ≥ 1 if and only if for all , ⊆ [] :  () − ( ∪ ) ≤   ∈ \ ( () −  ( ∪ {})) [33, Definition 2].Consider the following instance for BMMH.The input graph is a path of five nodes.All the nodes are red, except the middle node is blue.Let  = ∅ and  be the two edges that connect both red endpoints to the blue center.Then,  () −  ( ∪ ) > 0, but for all  ∈  \  :  () −  ( ∪ {}) = 0. □

ALGORITHMS FOR MINIMIZING AVERAGE HITTING TIME
We first discuss our algorithms for the BMAH problem, as we can use them as a building block for the BMMH problem.

A Greedy bicriteria algorithm
According to Observation 4 the objective function  of BMAH is supermodular.It also holds that  ≥ 1, as it takes at least one step to hit a blue node starting from any red node.There are two main ways for minimizing a non-negative monotone-decreasing supermodular function in the literature.One approach gives bounds that depend on the total curvature [49]  1− , for any  > 0 and curvature  ∈ [0, 1).However, we do not utilize this algorithm since it is not straightforward how to bound the curvature of , nor how to practically implement their algorithm.
Instead, we follow the bicriteria approach of Liberty and Sviridenko [32], where they show that the classic greedy algorithm gives a bicriteria (1 + )-approximation by allowing the addition of more than  elements.Their results are applicable if either the objective function is bounded away from zero (which holds in our case, since  ≥ 1), or one has an approximate initial solution.An immediate consequence of their results is Lemma 1.
Lemma 1.The Greedy strategy (Algorithm 1) returns a (1 + )approximation to an optimal solution of BMAH (that adds at most  edges) by adding at most ⌈ ln (  3  )⌉ edges.
Proof.We may assume  ≥ 1.Let  * denote an optimal solution to BMAH with at most  edges.We always have a crude approximate initial solution  0 = ∅, since for any  * it holds that (∅) ≤  3 ≤  3 ( * ).The first inequality holds because hitting times in connected graphs are upper bounded by  3 [34].The second inequality follows from  ≥ 1.The lemma essentially follows by application of the results of Liberty and Sviridenko [32,Theorem 6].We give a short proof here for completeness.Let   be the output of Algorithm 1, where  = ⌈ ln (  3  )⌉.By using the supermodularity and monotonicity of , we get the following recursion:

□
Note that finding the new edge that minimizes line 3 in Algorithm 1 does not require a search over O ( 2 ) possible non-edges.By Observation 1 we only need to choose a red endpoint, reducing the search to O () possible choices in each iteration.

Greedy+: Speeding up Greedy
In practice, lazy evaluation [41] will typically reduce the number of function evaluations that Algorithm 1 makes to .Nonetheless, function evaluations are still expensive.They require either solving a linear system of equations with O () variables, or directly computing the fundamental matrix (I − Q) −1 of an absorbing Markov chain with absorbing states , where Q denotes the probability transition matrix between the transient red nodes [27].
The high-level idea is to write  as a certain quadratic form of the fundamental matrix of an absorbing Markov chain with absorbing states being the blue nodes.The fundamental matrix of an absorbing Markov chain can be written as an infinite matrix power series [27].We will estimate this infinite series by a truncated sum.The truncated sum can be estimated efficiently by simulating random walks of bounded length, while the error induced by the truncation can also be shown to be small.The point of truncation will depend on certain graph parameters, such as the mixing time and average degree of the red nodes.Since most real-life graphs typically have bounded mixing time and bounded average degree, our approach is expected to work well in practice.This idea of truncating a matrix power series has been recently used by Peng et al. [46] and prior to that by Andoni et al. [3].
The following holds for any connected graph, so without loss of generality we prove our estimates for (∅).Let  = || throughout this proof.Let ℎ  ∈ R  ×1 denote the vector of expected hitting times from the red nodes until absorption by the blue nodes.Let Q ∈ R  × be the transition matrix between red nodes, obtained by deleting the rows and columns of corresponding blue nodes in the random-walk transition matrix P = D −1 A. It is well-known [27] that ℎ  = (I  −Q) −1   .Since (∅) is the average of ℎ  , we can write where the second equality follows from Lemma 2. Let ,  ∈ (0, 1).For every  ∈ , run  ≥ ℓ 2  2 ln( 2  ) absorbing random walks ( are the absorbing nodes) with bounded length ℓ and record how many steps are taken.Take the empirical average over these  trials, for every , and stack the results in a vector ĥ.With probability at least 1 − , ĥ ∈ (1 ± ) ℓ =0 Q    .Hence, also with probability at least 1 − , it holds that 1      ĥ ∈ (1 ± ) 1 .
Proof.We can give a probabilistic interpretation to  1 .We argue that it is the average of the hitting times-over all the red nodes as starting points-for absorbing random walks with bounded length ℓ.In other words, the walk halts when either it has taken more than ℓ steps, or it hits a blue node, whichever comes first.Consider such a bounded absorbing walk starting from  ∈ .Let   be the random variable counting the number of times the walk visits red node  ∈ .Let    be 1 if this walk is in state  after exactly  steps, and 0 otherwise.Clearly   = ℓ =0    .So the expectation satisfies Thus ℓ =0 Q    is a vector where the -th entry is the hitting time of an absorbing (with absorbing states ) bounded (with length ℓ) random walk starting from red node .And  1 is the average of this vector.To find a (1 ± )-estimation of  1 it suffices to find a (1 ± )estimation of each entry of ℓ =0 Q    .Now it is straightforward to estimate every -th entry of ℓ =0 Q    within a factor of (1 ± ).Run  walks from  and compute the empirical average of their lengths as the estimator.Such an estimation is exactly the topic of the RePBubLik paper [24, Lemma 4.3], and the claim in Lemma 2 follows easily from Hoeffding's inequality and the union bound.□ Next we show how to bound  2 as a small fraction of (∅), for a sufficiently large choice of ℓ.
Proof.Let D  ∈ R  × be a diagonal matrix with diagonal entries being the degrees of the red nodes in .All entries  1 , . . .,   are at least one, since  is assumed to be connected.Recall that   =  1 +...+   is the average of these entries.Let A  ∈ R  × be the adjacency matrix of is real-valued, symmetric, and similar to Q.So the real-valued eigenvalues  1 ≥  2 ≥ . . .≥   of M are also the eigenvalues of Q (by similarity) and M has a orthonormal eigendecomposition M = UΛU  .Let  = max{| 1 |, |  |} be the spectral radius of M (and of Q).By symmetry of M, we have that ∥M∥ 2 = .We know  < 1, since Q is a weakly-chained substochastic matrix, 1 [5, Corollary 2.6].Hence, Proof.We use the estimate ĝ = 1     ĥ from Lemma 2 as our estimate of (∅) with parameter  ′ = /2.Recall that (∅) =  1 + 2 .Since ĝ is an estimation of  1 by Lemma 2, it holds (w.h.p.) that ĝ , where the second to last inequality uses Lemma 3.
The time complexity follows from Lemma 2 by plugging in  = 1/  and the choice of ℓ from Lemma 3. We need to do this for every red node and there could be O () of them.Every bounded random walk also needs at most ℓ steps.□

Greedy on the estimated values.
Lemma 4 states that we can find good estimates ĝ(•) ∈ (1 ± )(•) relatively fast and with good probability.A natural question to ask is whether running the greedy bicriteria approach from Algorithm 1 on ĝ (instead of ) still ensures some form of approximation guarantee, given that ĝ is not guaranteed to be supermodular.A similar question has been answered affirmatively for the classic greedy algorithm of Nemhauser et al. [43] for maximizing a nonnegative monotone submodular function under a cardinality constraint, by the work of Horel and Singer [25].They showed that when  < 1/, the greedy still gives a constant-factor approximation [25, Theorem 5].
We show an analogue result for the bicriteria greedy approach [33] to minimization of our monotone decreasing nonnegative supermodular objective function , under a cardinality constraint.
Theorem 1. Algorithm 2 returns a set   that satisfies where OPT = () and  is an optimal solution of BMAH with at most  edges.

ALGORITHMS FOR MINIMIZING MAXIMUM HITTING TIME
Next we turn our attention to the BMMH problem.We first show that a -approximation for BMAH results in an (2|| 3/4 )-approximation for BMMH.We then give a completely different approach based on the asymmetric -center problem [26].

Relating the average and the maximum hitting times
The BMMH objective function  is not supermodular (Observation 4), although we know that for every  ∈  each individual hitting time  + (, ) is a supermodular and monotonically decreasing set function of  (Observation 2 and Observation 3).We exploit this idea, by upper bounding  by an expression that involves  (Theorem 2).Since also  ≤  , we can use any approximation algorithm for minimizing  (for instance, the methods in Sections 5.1 and 5.2) and use the upper bound to derive an approximation guarantee for BMMH.
Theorem 2 relates the average and maximum hitting time from red to blue nodes.We credit Yuval Peres [47] for the proof of Theorem 2, in case the set  is a singleton, and extend the proof here when  contains more than one node.Theorem 2 gives a remarkable non-trivial O ( 3/4 ) upper bound on the ratio of the maximum and average hitting times between two groups that constitute a bipartition of the graph.Theorem 3 shows that there exists graphs for which this is tight.In contrast, if we disregard the bipartition, and take the average and the maximum hitting time over all nodepairs instead, then it is known that this ratio can grow as bad as O (), but not worse [16].
Theorem 2. Given an undirected connected graph  with a valid bipartation  = {, }.Let  ⊆ ( × ) \  be a set of non-edges.Then, Proof.It clearly suffices to prove this for  = ∅.Since  is connected, the hitting time between any two nodes is finite-valued.The first inequality is immediate since the maximum is not less than the average.Consider the second inequality.Let  > 0 be some real number, which we will later choose carefully.Categorize the nodes into two groups, the ones with a "high" hitting time and the ones with a "low" hitting time to the blue nodes.More concretely, define and   as the complement of this set.Since it follows that |  | < ||/.The first inequality holds because   ⊆  (since  ⊆ ), and the second inequality uses the definition of   .Now consider the graph  * with |  | + 1 nodes, obtained by the following contraction in : All nodes in  are contracted to a single supernode .If  ∈   has several edges connecting it to  in , we only keep one of these edges.This edge connects  to  in  * .This contraction process has the property that for all  ∈   it holds   (, ) ≤   * (, ), which is easily proved by a straightforward coupling argument.Now we can bound   * (, ).Since  * is connected, contains no multi-edges, and has |  | + 1 nodes, it follows that   * (, ) ≤ |  | 3 by a well-known result (see, for instance, Lawler [29], and Brightwell and Winkler [9]).By the strong Markov property [31,Appendix A.3], it holds for all  ∈  : where the last inequality holds because |  | < ||/ and (∅) ≥ 1.Now Eq. ( 7) is minimized for the choice  = || 3/4 , and hence  (∅) ≤ 2|| 3/4 (∅).□ To see the tightness of the upper bound in Theorem 2, consider the following family of graphs.Theorem 3.There exists graphs for which the ratio  (∅)/(∅) ∈ Ω( 3/4 ).
Proof.Consider the following star-path-clique configuration.We have a star consisting of  nodes, with center node .Attach to  a ( 1/4 ,  1/4 )-lollipop graph (see Brightwell and Winkler [9]).More precisely, attach to  a path of  1/4 nodes and to the other endpoint of this path we attach a clique of also  1/4 nodes.Set  = { } and  =  \ .Clearly  (∅) ∈ Θ( 3/4 ) since a random walk starting from a clique node needs a cubic number of expected steps with respect to the size of the lollipop to reach  [9].On the other hand, and the claim follows.□ Putting the pieces together, we get the following result.

An asymmetric 𝑘-center approach
Hitting times in both undirected and directed graphs form a quasimetric, in the sense that they satisfy all requirements to be a metric except symmetry, since it generally holds that   (, ) ≠   (, ).However, hitting times do satisfy the triangle inequality [34].We will try to exploit this, by linking the BMMH problem to the following variant of the asymmetric -center problem.Problem 3 is a variant of the classic asymmetric -center problem [26].It is not hard to see that Problem 3 is as hard to approximate as the classic problem.Additionally, existing algorithms [4,44] for the classic problem can easily be modified to solve Problem 3.
We show a relationship between BMMH and Problem 3. Let I 1 be an instance to BMMH.Lemma 5 states that the optimum of Problem 3 on a well-chosen quasi-metric space I 2 (which depends on the instance I 1 ) is a lower bound for the optimum of BMMH on instance I 1 .We define this quasi-metric space in Definition 2. Definition 2 (modified hitting time qasi-metric).Let  = ( ∪ , ) be an input graph to BMMH.Associate with  a single new point , and consider function  : ( ∪ {}) × ( ∪ {}) → R ≥0 defined as Proposition 1. ( ∪ {}, ) is a quasi-metric space.
Proof.We prove the triangle inequality  (, ) ≤  (, ) +  (, ), as the identity axiom is satisfied.The triangle inequality holds when  ≠ , , , because of standard triangle inequality of hitting times.So consider the case where  is present in the inequality.There are three cases involving ,  ∈  and : • Proof.Consider the red endpoints  = { 1 , . . .,   } ⊆  of a set of edges  to BMMH that achieves the optimum value  * .Note that some endpoints might be the same, so  is a multiset.Let  * =  +  be the augmentation of  with  .
Create a feasible solution to Problem 3 by taking  as the set of chosen centers.We claim that for every  ∈  ∪ {} it holds that  (,  ∪ {}) ≤  * .We have three cases.If  = , the claim is trivially satisfied.If  ≠  and   (, ) ≤  * , the claim also holds, Input:  = ( ∪ , ), parameter  ≥ 1.
1: Run any -approximation algorithm [4,44] for asymmetric -center variant (Problem 3) on the quasi-metric space defined in Definition 2, with fixed center  = .Proof.By using a -approximation for Problem 3 and using Lemma 5 we get that for all  ∈  it is   (, ) ≤ min  ∈   (, ) ≤  * ≤  * .Consider a random walk that starts from any  ∈ .In expectation we need at most  * steps to encounter a node  ∈ .We know that every  ∈  is incident to some edge connecting it to .This is either a new shortcut edge  ∈  from Algorithm 3, or if we cannot add a new shortcut edge incident to , this implies that  is already connected to all the blue nodes.In either case, when the walk is at , we have a chance of at least 1  m +1 to hit the blue nodes in the next step.On the other hand, if the walk does not follow such an edge, the walk needs at most another  * steps in expectation to reach  again, and the process repeats.Of course, the walk might have encountered a blue node along the way, but this is only beneficial.So we conclude that for all  ∈ :  + (, ) ≤ ( * + 1) The last step follows by splitting the sum as a geometric series and an arithmetico-geometric series, which states that ∞ =1   =  (1− )2 for 0 <  < 1. □

EXPERIMENTAL EVALUATION
Although our work is mostly theoretical, we will compare the performance of our most scalable algorithm Greedy+ with several baselines on real-life datasets.In particular, we want to measure how Greedy+ reduces the objective functions of BMAH and BMMH when we add an incrementally larger set of shortcut edges to the graph, and we want to verify if other existing algorithms are effective for these two tasks.
Baselines.We will compare with the fastest variant of the RePBubLik algorithm, which is the RePBubLik+ algorithm [24], as well as with three simplified variants of RePBubLik+.We set the parochial nodes [24] of the algorithm equal to .The RePBubLik+ algorithm ranks the parochial nodes according to their random-walk centrality (only computed once), and includes a penalty factor that favours the insertion of new shortcut edges to red nodes that have not been shortcut before.The first variant, PureRandom [24] selects the endpoints of the new edges uniformly at random from  and .The second variant is Random Top- Central Nodes ( -RCN) [24], which sorts the top  red nodes in order of descending random-walk centrality and picks  source nodes uniformly from this set of  nodes.The third variant is Random Top- Weighted Central Nodes ( -RWCN), which accounts additionally accounts for the random-walk transition probabilities in the ordering [24].Finally, we compare with the ROV algorithm [19].ROV outputs  shortcut edges such that their addition aims to minimize the Random Walk Controversy (RWC) score [21] of the augmented graph.The RWC score tries to capture how well separated the two groups are with respect to a certain controversial topic.It considers candidate edges between high-degree vertices of both groups  and .Then the top- edges are picked with respect to the RWC score.All parameters of the aforementioned algorithms are set to their standard settings.Experimental setup.All experiments are performed on an Intel core i5 machine at 1.8 GHz with 16 GB RAM.Our methods are implemented in Python 3.8 and we made publicly available. 3We use the same datasets as RePBubLik [24], see Table 2. Implementation of Greedy+.We set  =  = 0.1 in our experiments.
In order to draw a fair comparison with the other baselines, we run Greedy+ for  steps and not more.We also simplify the way we calculate the average  over all the red nodes in each iteration.Lemma 2 states that we need to run several walks starting from each  ∈  and then compute the empirical average, but we only do this  for a randomly chosen subset of ||/10 red nodes.This significantly sped up the process, without sacrificing too much in quality.
Results. Figure 1 shows the performance of the proposed algorithms and baselines.On the -axis it shows the number of new shortcut edges  as a fraction of the total number of red nodes ||.We observe that our algorithm Greedy+ is competitive, performing very similar to RePBubLik+, for minimizing both objective functions  and .Both these algorithms outperform the other baselines.The slightly better performance for RePBubLik+ in some cases might be due to specific parameter settings, as well as the fact RePBubLik+ is repeated 10 times whereas our algorithm only once.
Intuitively RePBubLik+ and Greedy+ are in fact expected to perform very similar since they both are practical approximative versions (with different ways of estimating) of the same underlying greedy algorithm: picking the next edge that maximizes the RePBubLik+ objective function is theoretically the same edge that Greedy (see Algorithm 1) would select.We note that the y-axis shows the exact function evaluations of  and , which was done by solving a linear system with O () variables (see Sect. 5.2), which was relatively time consuming and an important reason why we restricted ourselves to smaller datasets.

CONCLUSION
In this paper we studied the problem of minimizing average hitting time and maximum hitting time between two disparate groups in a network by adding new edges between pairs of nodes.In contrast to previous methods that modify the objective so that it becomes a submodular function and the optimization becomes straightforward, we minimize hitting time directly.Our approach leads to having a more natural objective at the cost of a more challenging optimization problem.For the two problems we define we present several observations and new ideas that lead to novel algorithms with provable approximation guarantees.For average hitting time we show that the objective is supermodular and we apply a known bicriteria greedy method; furthermore, we show how to efficiently approximate the computation of the greedy step by sampling bounded-length random walks.For maximum hitting time, we show that it relates to average hitting time, and thus, we can reuse the greedy method.In addition, we also demonstrate a connection with the asymmetric -center problem.

Theorem 4 .
∈   (, ) ≤   (,  * ) ≤   * (,  | walk uses a new edge incident to  * ) ≤  * .□Lemma 5 allows us to analyze Algorithm 3. We write  m = max  ∈   () for the maximum degree of the nodes in , and  for the approximation guarantee of an asymmetric -center algorithm.The state-of-the-art approximation for asymmetric -center is  = O (log * ()) by two different algorithms of Archer[4].Algorithm 3 is an O (  m )-approximation for BMMH.

Figure 1 :
Figure 1: The y-axis shows the performance of the algorithms in reducing the functions  (first row) and  (second row) in terms of the number of newly added shortcut edges .The x-axis shows the number of new edges  as a fraction of ||.

Table 1 :
Results summary.The approximation factor is computed with respect to an optimal solution that uses at most  edges. m is at most the maximum degree (see Section 6.2).

Table 2 :
Datasets used in the experiments.For each network, we extracted the largest connected component.|| ↔ denotes the number of inter-group edges, and ||  the total number of edges.