Multiplex Conductance and Gossip Based Information Spreading in Multiplex Networks

Yufan Huang and Huaiyu Dai Department of Electrical and Computer Engineering North Carolina State University, USA Email: {yhuang20, [email protected]}

Abstract—In this work, we study the information spreading simplified version of multilayer networks already captures time in multiplex networks, adopting the gossip (random-walk) many interesting multi-scale and multi-component features, based information spreading model. A new metric called mul- and serves as a good starting point for our intended study. tiplex conductance is defined based on the multiplex network structure and used to quantify the information spreading time in There has been some works on information spreading in a general multiplex network in the idealized setting. Multiplex multiplex networks, which are based on the compartmental conductance is then evaluated for some interesting multiplex epidemic spreading models [3, 4], and mainly focus on the networks to facilitate understanding in this new area. Finally, macroscopic network behavior. Our work instead adopts the the tradeoff between the information spreading efficiency im- gossip (random-walk) based information spreading model [5], provement and the layer cost is examined to explain the user’s social behavior and motivate effective multiplex network designs. which is considered as reflecting more details of the underlying communication dynamics and network structures, and can facilitate the quantification of the information spreading time. I.INTRODUCTION To the best of our knowledge, this information spreading In the past election year, one of the most important tasks model has not been explored for multiplex networks. for presidential candidates is to disseminate their words In this work, the gossip-based information spreading time and opinions to voters in a fast and effective manner. The is found to be closely connected to a newly defined metric underlying research problem on information spreading has multiplex conductance. Specifically, our contributions can be already received great interest and been extensively studied in summarized as follows: a single network. However, with the continuous advancement • A new metric, multiplex conductance Φmp, is defined of modern technology, the ways that the candidates can based on the multiplex network structure, and it is shown exploit to promote their influence are no longer limited to −1 that Θ(Φmp · log n) is a good estimate for information the campaign tour; radio networks, TV networks, telephone spreading time in many multiplex networks of interest. networks, and Internet have all been utilized for their pur- • Multiplex conductance of some interesting multiplex poses. Especially, with the phenomenal popularity of social networks is evaluated to shed light on this burgeoning networks, all candidates have utilized their Facebook and research field. Twitter accounts to post and spread their political agenda, • The tradeoff between the cost of additional layers and through which their words can be shared and disseminated in the improvement of information spreading efficiency is an unprecedented range and scale. Therefore, with increasingly discussed from both the user’s and the network designer’s complicated interconnections and interactions, various kinds aspect. of communication networks and social media have formed The rest of the paper is organized as follows. The system a new network structure that enables people to spread and model and gossip algorithm for multiplex networks are intro- receive information simultaneously through multiple channels duced in Section II. Section III presents the main theoretical and platforms. Recently, multilayer network models have been results for information spreading time in multiplex networks introduced to facilitate relevant studies on emerging inter- and the evaluation of multiplex conductance. In Section IV, connected complex networks [1, 2]. In this work, we take a some discussion on the trade-off between the layer cost and first step to investigate information spreading in a special type the improvement of information spreading efficiency is given. of multilayer networks, termed multiplex networks, for which Section V concludes the work. all layers share the same set of nodes. In practice, the same set of nodes may correspond to individuals who can communicate II.PROBLEM FORMULATION through multiple networks or platforms, and duplicates of the In this section we briefly introduce the network and system same node may represent different communication devices or models. More details can be found in the technical report [6]. social accounts a person may have. Arguably, this somewhat A. Basic Models This work was supported in part by National Science Foundation under Grants ECCS-1307949 and EARS-1444009, and in part by Army Research 1) Multiplex Network: A multilayer network is modeled by a M Office under Grant W911NF-17-1-0087. family of graphs {Gm , (Vm,Em)}m=1 that constitute the layers of this complex system, together with the interlayer unique nodes connected to u in any layer, i.e., v ∈ Neg(u) M connections represented by Eαβ, for any two different layers S if (u, v) ∈ Em. If v ∈ Neg(u), the link (u, v)’s existing Gα and Gβ. In this study, we will focus on multiplex networks, m=1 1 for which all layers share the same set of nodes, i.e., V1 = layer set is defined as L(u,v) = {α;(u, v) ∈ Eα}, and the V2 = .... = V = [n], and interlayer connections exist only corresponding (u, v) link at layer α is denoted as (u, v)α. between the duplicates of the same node at different layers, i.e., Eαβ = {(vα, vβ); v ∈ V } for all α 6= β, where vα is the A. Idealized Information Spreading Time duplicate of node v in layer α. The analysis for information spreading in the multiplex net- 2) Synchronous Time Model: In this work, the synchronous work is mainly complicated by two factors: overlapping edges time model is adopted, i.e., all nodes in the network take action among layers and heterogeneous contacting probabilities at simultaneously at discrete time steps. This is a common model different layers. They render the exact estimation of infor- used for studying gossip-based information spreading [8]. mation spreading time in the multiplex network intractable. 3) Gossip Algorithm in Multiplex Network: For the gossip Therefore, an idealized setting is considered in the following algorithm in the single network, in each time slot, each node so that the corresponding information spreading time can contacts one of its neighbors independently and uniformly at be analyzed, which serves as a good lower bound for the random. The push-pull model is considered for transmitting information spreading time of the original multiplex network. information. Specifically, in each round, for the push opera- First, to handle the overlapping edges among layers, an tion, every informed node randomly chooses a neighbor and aggregated multigraph representation for a multiplex network attempts to pass the information, while for the pull operation, is constructed as follows: every uninformed node randomly chooses a neighbor and Definition 2: Given a multiplex network G = {Gm , attempts to grab the information. In this work, we will consider M ˜ (V,Em)}m=1, the corresponding aggregated multigraph G = the following gossip algorithm for a multiplex network: Before ˜ ˜ ˜ ˜ M (V, E) is defined such that V = V and E = ]m=1Em , the gossip process, it is assumed that all duplicates of the same 2 {(u, v)α; u ∈ V, v ∈ Neg(u), α ∈ L(u,v)}, where ] stands node are synchronized. During a gossip step, all nodes and for the non-unique set union, i.e., the same links at different their duplicates contact one of their neighbors uniformly at layers are all kept. random in all layers simultaneously. After each gossip step, To get around heterogeneous gossip at different layers, node the newly informed nodes (if they exist) will broadcast the u’s contacting probability for each link (u, v)α is unified information to all their duplicates. 1 as (u) = ((u, v)α) = (denoted as link P P min dm(u) m∈{1,...,M} B. Information Spreading Time picking probability). This over-optimistic choice simplifies our The metric commonly used to measure the efficiency of gossip analysis, while still providing a good lower bound as shown based information spreading is the information spreading time. below. Denote St as the informed node set at round t, with S0 = The idealized setting is formed through a uniform gossip {s}, for some arbitrary s ∈ V . The information spreading with link picking probability P(u), ∀u ∈ V , on the ag- ˜ time in a network G of size n, Tspr(G, γ), γ > 0, is modeled gregated multigraph G constructed above. The information as the stopping time by which all nodes are informed with spreading time in this idealized setting for an arbitrary multi- −γ probability 1 − O(n ) [8], i.e., Tspr(G, γ) = sup inf{t : plex network is quantified below. s∈V −γ Definition 3: The multiplex conductance Φmp of a multiplex P r(St 6= V |S0 = {s}) ≤ O(n )}. M network {Gm , (V,Em)}m=1 is defined as III.MAIN RESULTS |cutT (S, V − S)| For the gossip model, analyzing the information spreading Φmp = min M , (1) process is difficult even in a single network due to the S⊂V,volT (S)≤|E|T volT (S) heterogeneous network topology and random gossip processes. M M P P The multiplex network structure introduces interconnections where volT (S) = volm(S), |E|T = |Em|, and m=1 m=1 and interactions among layers, which further complicate the M P analysis. In this study, we slightly relax the problem and |cutT (S, V − S)| = |cutm(S, V − S)|. volm(S) is the m=1 endeavor to find the information spreading time in a general degree sum of all nodes in the node set S at layer m (volume), multiplex network in an idealized setting. and |cutm(S, V − S)| is the number of edges between node First, the following definition is needed for the following set S and V − S at layer m. analysis: Theorem 1: For an M-layer multiplex network with n nodes, Definition 1: Given a multiplex network G = {Gm , the information spreading time in the idealized setting is at M (V,Em)}m=1, for each node u ∈ V , dm(u) is defined as −1 1 most 200(γ+2)Φmp·(log n+ log M) rounds with probability the node degree of u in layer m. The maximum total node 2 M P 1 degree is then defined as ∆max = max( dm(u)). The total Link (u, v) may exist in several layers simultaneously. u∈V m=1 2Note that multiple edges are allowed between a pair of nodes in a neighbor set Neg(u) of node u is defined as the set of all multigraph. −γ 1 − O(n ), where Φmp is the multiplex conductance of this spreading process in this stage into phases each comprised multiplex network. of 5∆max/(dΦmpvolT (S0)e) rounds. A phase is considered First, the following sequence of random variables solely successful if the total volume of the informed node set at related to the pull operation will be introduced to facilitate the end of that phase is at least ∆max. Therefore, in the our analysis. Let L1,L2, ... be a sequence of random variables first d2γ ln ne phases, the probability that none of them is d2γ ln ne −γ with Li (i ≥ 1) defined as follows. We distinguish two cases: successful is at most (1 − 1/2) ≤ O(n ). Therefore, • If volT (Si−1) ≤ |E|T , then by Definition 3, with at most t = 3γ ln n · 5∆max/(dΦmpvolT (S0)e) rounds, 1 the total volume of the informed node set has volT (St) ≥ |cutT (Si−1,Ui−1)| ≥ M dΦmpvolT (Si−1)e ≥ 1 −γ ∆max with probability at least 1 − O(n ). M dΦmpvolT (S0)e, where Ui−1 = V − Si−1 is the set of uninformed nodes at round i − 1. Let In the second stage, by Lemma 1(c), if ∆max ≤ volT (S0) ≤ R = dΦmpvolT (S0)e and Ei be an arbitrary subset of |E|T , then with probability at least 1/2, it takes at most 1 M d4/Φmpe rounds until the total volume of the informed node ]m=1cutm(Si−1,Ui−1) consisting of M R edges. Set Ei is (arbitrarily) fixed at the beginning of round i–before set is increased to at least min{2volT (S0), |E|T + 1}. Sim- the round is executed. Define the minimum volume of a ilarly, divide the information spreading process in this stage node u as M · min{dm(u)}. For each node u ∈ Ui−1, let into phases of d4/Φmpe rounds each. A phase is successful if m the total volume of the informed node set at the end of the δi,u be the 0/1 random variable with δi,u = 1 if and only if in round i node u pulls the information through some phase is at least min{2volT (Si), |E|T + 1}, where Si is the P set of informed nodes at the beginning of that phase. Then, edge in Ei. Then, Li = (δi,uM · min{dm(u)}). u∈Ui−1 m for any k, the probability that the k-th phase is successful • If vol (S ) > |E| , then L = R. T i−1 T i is at least 1/2, regardless of the outcome of the previous Then, the following two lemmas are needed to prove k − 1 phases. Let B(k, 1/2) denote the binomial random Theorem 1: variable for k trials each with success probability of 1/2. Lemma 1: Then, by the Chernoff bound, the probability that fewer than P P (a) E[ k≤i Lk] = iR and V ar( k≤i Lk) ≤ iR∆max, η = log |E|T of the k = (2γ + 4)η phases are successful 2 where R = dΦmpvolT (S0)e. is at most P r(B(k, 1/2) ≤ η) ≤ e−2(η−k/2) /k ≤ O(n−γ ), (b) If volT (S0) < ∆max, then P r(volT (Si) ≥ ∆max) ≥ since |E|T ≥ n − 1. And since at most η successful phases 1/2, for i ≥ 4∆max/(dΦmpvolT (S0)e). are required for the total volume of the informed node set −γ (c) If ∆max ≤ volT (S0) ≤ |E|T then P r(volT (Si) ≥ exceeding |E|T , it follows that with probability 1 − O(n ) min{2volT (S0), |E|T + 1}) ≥ 1/2, for i ≥ 4/Φmp. the number of rounds required for the same goal is at most (d) If volT (S0) > |E|T then P r(volT (Ui) ≤ volT (U0)/2) ≥ k · 5/Φmp ≤ (2γ + 4)(2 log n + log M)(5/Φmp) as |E|T ≤ 1/2, for i ≥ 6/Φmp, where Ui = V − Si is the set of M · n2. uninformed nodes at round i. Finally, by Lemma 1(d), if volT (Si) > |E|T , by similar Lemma 2: ([8]) Let εPUSH (a, b, t) denote the event that reasoning as before, we can show that once the total volume the push operation spreads to node b the information started of informed nodes has exceeded |E|T , then (2γ +4)(2 log n+ at node a in at most t rounds; and εP ULL(b, a, t) denote log M)(7/Φmp) rounds suffice to inform all nodes with the event that the pull operation spreads to node a the probability 1 − O(n−γ ). information started at node b in at most t rounds. Then Combining the above three cases and applying the union −γ P r(εPUSH (a, b, t)) = P r(εP ULL(b, a, t)). bound, we obtain that, with probability 1 − O(n ), all 1 −1 Remark 1: The random variables Li are used to approximate nodes get informed within 50(γ +2)(log n+ 2 log M)(Φmp + the increment of volT (Si). With the expectation and variance ∆max/dΦmpvolT (S0)e) rounds given any initial informed of the sum of random variables Li in Lemma 1(a), different node set S0 when only the pull operation is considered. increasing speeds of volT (Si) are shown in Lemma 1(b), Then let vmax be the node of maximum total node degree (c), and (d) for three different stages ((d) actually shows ∆max (see Definition 1). From above, the pull operation the decreasing speed of its complement volT (Ui)). Lemma distributes the information from vmax to any other nodes in −1 1 2 shows the symmetry between the pull and push operation. 100(γ + 2)Φmp · (log n + 2 log M) rounds w.h.p. Then by The proof of Lemma 1 (and more details for that of Theorem Lemma 2, the push operation can spread to vmax the infor- −1 1) can be found in [6]. mation started at an arbitrary source node s in 100(γ+2)Φmp · 1 Sketch of Proof for Theorem 1: When only the pull operation (log n + 2 log M) rounds with the same high probability. is considered, it can be seen from Lemma 1 that the total Therefore, the push-pull operation can spread the information −1 1 volume of the informed node set volT (Si) increases in started at s to all nodes in 200(γ + 2)Φmp · (log n + 2 log M) different ways in the three different stages. The informa- rounds w.h.p.  −1 tion spreading analysis is given below accordingly. In the Remark 2: By Theorem 1, we have shown that Θ(Φmp · first stage, by Lemma 1(b), if given volT (S0) < ∆max, log n) is a good estimate for information spreading time in a after at most 5∆max/(dΦmpvolT (S0)e) rounds, the total general multiplex network in the idealized setting. It becomes volume of the informed node set becomes at least ∆max a true lower bound for the actual information spreading time with probability at least 1/2. Now, divide the information when the layer number of multiplex networks is sufficiently 4 M−Ring Multiplex Networks large, as shown below. x 10 7 Theorem 2: Given an M-layer multiplex network G = 1−Layer (Simulation) 2−Layer (Simulation) M {G } with n nodes, there always exists a constant c > 0 6 2−Layer (Idealized) m m=1 n 2−Layer (Corrected) −1 3−Layer (Simulation) such that, when M > cn, Θ(Φmp · log n) is a lower bound of 5 3−Layer (Idealized) the actual information spreading time. 3−Layer (Corrected) 4−Layer (Simulation) Proof: See [6]. 4 4−Layer (Idealized) 4−Layer (Corrected) While Theorem 2 is interesting in theory, in practice, M is often a modest number. In this case, if it can be 3 further shown that the information spreading capability of 2 Information Spreading Time each layer is accurately measured (in the order sense) by the −1 1 corresponding conductance, Θ(Φmp ·log n) can still be a good lower bound for the actual information spreading time of a 0 0 2 4 6 8 10 4 multiplex network. More discussions can be found in [6]. For Network Size x 10 single networks, many graph models of interest actually satisfy Fig. 3: Information Spreading in Multiplex Networks with Identical this property. Prominent examples include complete graphs, Ring Graphs ring graphs, random geometric graphs (RGGs), and expander graphs (which include Preferential Attachment (PA) graphs as a special case). In this case, the maximum improvement common misconception that the best achievable improvement of information spreading efficiency from a single network for information spreading in a multiplex network formed with with conductance Φ to a multiplex network with multiplex similar-topology layers would be on the order of M. Our Φmp conductance Φmp can be orderly determined by PI = Φ . first example below dispels this misconception through an innovative design exploiting the second factor above.

350 7000 Simulation Ring-PA (Simulation) 1) Proposed Ring-Ring Multiplex Network: In this exam- Φm Ring-Complete (Simulation) 300 Φ 6000 (n) ple, a novel ring-ring multiplex network structure is proposed 250 5000 to demonstrate the impact of increased contacting opportu- 200 4000 nities in a multiplex network, for which an improvement as 3000 √ 150 large as Θ( n) can be achieved, as shown in Fig. 1. In 100 2000 Efficiency Improvement Efficiency Improvement this proposed multiplex network structure, the first layer is 50 1000

0 a normal ring, where nodes are identified with the integers 0 1 2 3 4 5 10 10 10 10 10 0 2 4 6 8 10 Network Size Network Size 104 1, 2, ..., n. Assuming without loss of generality that n +√ 1 is Fig. 1: Proposed Ring-Ring Fig. 2: Ring-Complete, PA not a prime number, find out its factor r that is closest to n. Then in the second layer, node with ID i is connected with two B. Multiplex Conductance Evaluation and Performance Im- nodes with IDs ((i+r) mod n) and ((i−r) mod n) instead provement (see Fig. 10 in [6] for details). It can be shown that the second layer is guaranteed with a ring structure. Then the multiplex Our main result above, as a generalization and advance of the conductance of this network structure is given by Φ = current art of the single network study, nicely encapsulates mp 2 min{r+1,(n+1)/r+1} . Detailed calculation can be found in the gossip-based information spreading performance in a n [6]. As shown in Fig. 1, the predicted information spreading multiplex network into a metric, its multiplex conductance. efficiency improvement P = Φmp = min{r + 1, n+1 + 1} is In this part, the multiplex conductance is evaluated for some I Φ r quite tight in this scenario. interesting multiplex networks to facilitate understanding. Remark 3: As can be seen from Definition 3, the multiplex conductance The second layer of the proposed ring-ring struc- is calculated through jointly minimizing the overall cut- ture dramatically increases the size of cut through rewiring volume ratio of the multiplex network. Evaluation of the of nodes without increasing the corresponding set volumes. single network conductance is in general a hard problem This design leads to an information spreading efficiency min{r+1, (n+1)/r+1} already, and existing results are mainly obtained for graphs improvement√ on the order of , which n with certain nice properties and usually in the order form [5]. is close to in most cases. The existence of multiple layers and interactions among layers 2) Different-Topology Multiplex Network: This example re- further complicates the evaluation. Therefore, as a first work in veals increased contacting opportunities in a multiplex network this area, we will focus on evaluating some multiplex networks from a new perspective. Note that multiplex networks don’t with special structures, which allow us to obtain simple need to be constrained to constructions of similar-topology and inspiring results. Based on the definition of multiplex layers. Intuitively, a marriage of different topologies can lead conductance, two key factors contribute to higher information to a dramatic improvement in information spreading for the spreading efficiency in multiplex networks as compared to the disadvantaged layer, as shown by the following two cases. single network: 1) availability of multiple channels M, and 2) • Ring-Complete Coupling: If a ring graph is coupled larger contacting opportunities |cutT (S,V −S)| . There may be a with another complete graph to form a ring-complete volT (S) Two- Ring- Proposed multiplex network, the corresponding multiplex conduc- Identical- Ring-PA Complete Ring-Ring tance is given by Φmp = Θ(1). Then the performance Ring Φmp User average: Θ(n) √ improvement is PI = Φ = Θ(n). Θ(1) Θ(1) √ Θ( n) (RCU ) minimum: Θ( n) • Ring-PA Coupling: Similarly, the corresponding multi- Network √ plex conductance of a ring-PA multiplex network is given Designer Θ(1) Θ(1) Θ(n) Θ( n) RC by Φmp = Θ(1). Detailed calculation can be found in ( N ) [6]. Again the performance improvement is Θ(n). Both TABLE I: Performance-Cost Tradeoffs results are verified by simulations in Fig. 2. 3) Identical-Topology Multiplex Network: Finally, the im- Remark 4: From the results in Table I, it can be seen that pact of layer number M is explored, considering a special overall the ring-PA coupling is the most effective multiplex type of multiplex networks formed with M layers of identical structure as far as information spreading is concerned. Since topology. When M identical graphs having conductance Φ the PA graph is a popular model for social platforms, this form a multiplex network, the overall multiplex conductance result partially reveals the beneficial impact of utilizing social can be shown as Φmp = MΦ. Therefore, an improvement networks for information distribution. of order M is expected for information spreading efficiency On the other hand, the preferential attachment design is in this setting, which is generally an over-estimate, as the much more complicated as compared with our proposed ring- effect of link collision at different layers is ignored. This ring structure. Therefore, as far as the overall network design effect is most severe for the identical-layer structure, and can is concerned (instead of utilizing existing network structures), be partially corrected by considering the average meaningful our proposed structure provides an insight for a careful contact for each node, which is upper bounded by ∆(1 − planning of the new layer structure that can greatly improve the 1 M overall efficiency for information spreading without incurring (1 − ∆ ) ), where ∆ is the largest degree of each layer. This simple correction leads to an improvement as shown in Fig. much additional cost. 3. For multiplex networks constructed by independent layers, V. CONCLUSIONAND FUTURE WORK the over-estimation error is usually not a concern. In this work, the gossip based information spreading is studied in multiplex networks. By defining the new metric multiplex IV. LAYER COSTAND INFORMATION SPREADING conductance Φ , Θ(Φ−1 · log n) is found to be a good EFFICIENCY mp mp estimate for information spreading in many multiplex networks In real life, the adoption of a new layer comes with an of interest. The multiplex conductance is then evaluated for additional cost. In this section, the tradeoff between the several interesting multiplex networks to help understand improvement of information spreading efficiency and the information spreading potentials of multiplex networks. By additional layer cost is discussed. In particular, we will exam further taking the additional layer cost into consideration, the two types of cost below.3 tradeoff between the cost and the information spreading effi- • Network Cost for a User: The cost of additional layers ciency improvement is discussed from both the user’s and the from a user u’s aspect is measured by the total degree network designer’s perspective to facilitate the understanding PM in this burgeoning area. of this node in all layers, i.e., CML(u) = m=1 dm(u). Therefore, the corresponding cost increase for the user u EFERENCES PM R m=1 dm(u) is CI,U (u) = d(u) , where d(u) is the node degree [1] M. Kivela,¨ A. Arenas, M. Barthelemy, J. P. Gleeson, Y. Moreno, and of u in the initial single network. M. A. Porter, “Multilayer networks,” Journal of Complex Networks, vol. 2, • Network Cost for the Network Designer: Different from no. 3, pp. 203–271, 2014. [2] S. Boccaletti, G. Bianconi, R. Criado, C. I. Del Genio, J. Gomez-´ the user’s aspect, for a network designer, the cost of a Gardenes,˜ M. Romance, I. Sendina-Nadal, Z. Wang, and M. Zanin, “The new layer is better measured by the number of total edges structure and dynamics of multilayer networks,” Physics Reports, vol. of it. Then the cost increase for the network designer is 544, no. 1, pp. 1–122, 2014. PM [3] C. Granell, S. Gomez,´ and A. Arenas, “Dynamical interplay between m=1 |Em| CI,N = |E| , where |E| is the number of total awareness and epidemic spreading in multiplex networks,” Physical edges of the initial single network. review letters, vol. 111, no. 12, p. 128701, 2013. [4] D. Zhao, L. Li, H. Peng, Q. Luo, and Y. Yang, “Multiple routes With the cost increase CI for each multiplex network, and transmitted epidemics on multiplex networks,” Physics Letters A, vol. the corresponding information spreading efficiency improve- 378, no. 10, pp. 770–776, 2014. PI [5] D. Shah, Gossip algorithms. Now Publishers Inc, 2009. ment PI , the reward-cost ratio can be defined as RC = . In CI [6] Y. Huang and H. Dai, “Multiplex conductance and gossip based Table I, given an initial single ring graph,4 tradeoffs between information spreading in multiplex networks,” NC State University, the performance improvement and the cost are evaluated for Tech. Rep., 2016. [Online]. Available: http://www4.ncsu.edu/%7Ehdai/ Multiplex%2DConductance%5FTP.pdf four related multiplex networks. [7] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, “Randomized gossip algorithms,” IEEE/ACM Transactions on Networking (TON), vol. 14, 3The choices of cost in this paper mainly serve to facilitate relevant no. SI, pp. 2508–2530, 2006. discussion. In practice, other meaningful costs may also be considered. [8] G. Giakkoupis, “Tight bounds for rumor spreading in graphs of a given 4A single ring graph may serve as a coarse representation of a highly conductance,” in Symposium on Theoretical Aspects of Computer Science localized network, severely limited in information spreading capability. (STACS2011), vol. 9, 2011, pp. 57–68.