Multiplex Conductance and Gossip Based Information Spreading in Multiplex Networks

1 Multiplex Conductance and Gossip Based Information Spreading in Multiplex Networks

Yufan Huang, Student Member, IEEE, and Huaiyu Dai, Fellow, IEEE

Abstract—In this network era, not only people are connected, different networks are also coupled through various interconnections. This kind of network of networks, or multilayer networks, has attracted research interest recently, and many beneficial features have been discovered. However, quantitative study of information spreading in such networks is essentially lacking. Despite some existing results in single networks, the layer heterogeneity and complicated interconnections among the layers make the study of information spreading in this type of networks challenging. In this work, we study the information spreading time in multiplex networks, adopting the gossip (random-walk) based information spreading model. A new metric called multiplex conductance is defined based on the multiplex network structure and used to quantify the information spreading time in a general multiplex network in the idealized setting. Multiplex conductance is then evaluated for some interesting multiplex networks to facilitate understanding in this new area. Finally, the tradeoff between the information spreading efficiency improvement and the layer cost is examined to explain the user’s social behavior and motivate effective multiplex network designs.

Index Terms—Information spreading, multiplex networks, gossip algorithm, multiplex conductance !

1 INTRODUCTION

N the election year, one of the most important tasks Arguably, this somewhat simplified version of multilayer I for presidential candidates is to disseminate their words networks already captures many interesting multi-scale and and opinions to voters in a fast and efficient manner. multi-component features, and serves as a good starting The underlying research problem on information spreading point for our intended study. has already received great interest and been extensively As a common tool for studying information spreading studied in a single network. However, with the continuous in the single network, the compartmental epidemic model advancement of modern technology, the ways that the (e.g., Susceptible-Infected-Removed (SIR) or Susceptible- candidates can exploit to promote their influence are no Infected-Susceptible (SIS) model) has been utilized to dis- longer limited to the campaign tour; radio networks, TV cover how the multiplex network structure affect the networks, telephone networks, and Internet have all been information spreading process [6–11]. It can be used to utilized for their purposes. Especially, with the phenomenal study a spreading process happening on all layers of a popularity of social networks, all candidates have utilized multiplex network. Cozzo et al. [9] proposed a contact-based their Facebook and Twitter accounts to post and spread their epidemic-like information spreading model in multiplex political agenda, through which their words can be shared networks. It further demonstrated that the critical point for and disseminated in an unprecedented range and scale. the network having a constant portion of informed nodes is Therefore, with increasingly complicated interconnections determined by the layer whose contacting probability ma- and interactions, various kinds of communication networks trix has the largest eigenvalue. In [7], Zhao et al. considered and social media have formed a new network structure a spreading process on a two-layer multiplex network with that enables people to spread and receive information a certain similarity between these two layers and showed simultaneously through multiple channels and platforms. that a positive degree-degree correlation between two layers Recently, multilayer network models have been introduced may lead to a smaller infection size in the end. Spreading to facilitate relevant studies on emerging inter-connected not only exists on intra-layer links but also on inter-layer complex networks [2–5]. In this work, we take a first step ones. The effect of the layer-switching cost on spreading to investigate information spreading in a special type of processes have been studied in [8], where different effective multilayer networks, termed multiplex networks, for which infection rates in the SIR model are considered for intra- all layers share the same set of nodes. In practice, the and inter-layer links. It is shown that when both layers same set of nodes may correspond to individuals who can have the same average degree, a lower layer-switching cost, communicate through multiple networks or platforms, and which is determined by the difference between intra- and duplicates of the same node may represent different com- inter-layer infection rates, can trigger the epidemics more munication devices or social accounts a person may have. easily (lower the epidemic threshold) while it suppresses the final epidemic size. Multiple spreading processes can also be addressed simultaneously through multiplex network • Yufan Huang and Huaiyu Dai are with the Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC, modeling. In [6, 10], Granell et al. studied the coupled 27695. processes of awareness and infection on multiplex networks. • Part of this work has been presented at 2017 IEEE International By adopting the heterogeneous mean-field approximation, Symposium on Information Theory (ISIT), Achen, Germany [1]. they quantified the effect of using the information spreading 2 process to stop the epidemic process, where the immunolog- is discussed from both the user’s and the network ical information spreading process in the social network will designer’s aspect. help deter the epidemic spreading process happening in the The rest of the paper is organized as follows. The system physical world. Wei et al. in [11] instead focused on two model, the gossip algorithm, and the definition of informa- cooperative spreading processes on a multiplex network tion spreading time for multiplex networks are introduced and proved that the cooperation can promote the spreading in Section 2. Section 3 presents the main theoretical results progress. for information spreading time in multiplex networks and However, the compartmental epidemic model mainly the evaluation of multiplex conductance. In Section 4, some focuses on the macroscopic network behavior. More specif- discussion on the trade-off between the layer cost and the ically, information is diffused from any informed/infected improvement of information spreading efficiency is given. node to its neighbors in a broadcasting-like manner; the Section 5 concludes the work. independent cascade model and linear threshold model widely used in the study of social networks [12] also belong to this category. This type of models is predominantly 2 PROBLEM FORMULATION used to describe group behaviors and determine the final In this section we introduce the network and system models size of informed nodes. However, this class of models is as well as the definition of information spreading time. by no means the only one considered in literature. First, as a tradeoff for mathematical elegance, some simplified 2.1 Basic Models assumptions are made about node behaviors and the underlying communication structure. In practice, broadcasting 1) Multiplex Network: A multilayer network is modeled by a , M is not always adopted for information spreading due to family of graphs {Gm (Vm,Em)}m=1 that constitute the various concerns including excessive resource consump- layers of this complex system, together with the interlayer tion and privacy. Actually, in many scenarios, one-to-one connections represented by Eαβ, for any two different layers communications happen naturally between two individuals Gα and Gβ. In this study, we will focus on multiplex through phone calls, text messages, emails, and other di- networks, for which all layers share the same set of nodes, i.e., rected communications. This type of information spreading V1 = V2 = .... = V = [n], and interlayer connections exist is best captured by another popular model, random-walk only between the duplicates of the same node at different based model, or gossip model [13]. The gossip based layers, i.e., Eαβ = {(vα, vβ); v ∈ V } for all α 6= β, where vα algorithms and designs possess many advantages: simple is the duplicate of node v in layer α. in implementation, efficient in resource utilization, and 2) Synchronous Time Model: In this work, the synchronous robust to network dynamics, thus becoming an appealing time model is adopted, i.e., all nodes in the network take architectural solution for many problems in large-scale action simultaneously at discrete time steps. This is a dynamic networks. One particular advantage of the gossip common model used for studying gossip-based information model, as compared to the aforementioned information spreading [19]. diffusion models, is to allow the quantitative study of the 3) Gossip Algorithm in Multiplex Network: For the gossip information spreading speed. In summary, our work adopts algorithm in the single network, in each time slot, each node the gossip (random-walk) based information spreading contacts one of its neighbors independently and uniformly model, which is considered as better capturing the personal at random. During each meaningful contact (for which communications behaviors in many scenarios, reflecting exactly one node has the piece of information1), the message more details of the underlying communication dynamics is successfully delivered in either direction (through the and network structures, and can facilitate the quantification “push” or “pull” operation). Specifically, in each round, for of the information spreading time. To the best of our the push operation, every informed node randomly chooses knowledge [14–18], the gossip based information spreading a neighbor and attempts to pass the information, while model has not been explored for multiplex networks. for the pull operation, every uninformed node randomly In this work, the gossip-based information spreading chooses a neighbor and attempts to grab the information. time is found to be closely connected to a newly defined There are two key differences for information spreading metric, multiplex conductance. Specifically, our contributions in a multiplex network: First, the message can be spread can be summarized as follows: on multiple layers simultaneously. Second, when a node • The informed probability model is proposed to facilitate gets informed at one layer, it will automatically become the study of information spreading in multiplex net- informed at all other layers. While these two assumptions works. are somewhat idealistic, we will use them in this study to • A new metric, multiplex conductance, Φmp is defined explore the maximum potential for information spreading based on the multiplex network structure, and it is in multiplex networks. In particular, we will consider the −1 shown that Θ(Φmp · log n) is a good lower bound following gossip algorithm for a multiplex network: Before for information spreading time in a general multiplex the gossip process, it is assumed that all duplicates of the network when the layer number is large. same node are synchronized. During a gossip step, each • Multiplex conductance of some interesting multiplex node and its duplicates contact one of its neighbors (not networks is evaluated to shed light on this burgeoning 1 research field. While that the classic single piece information spreading problem is focused in this work, most of our results can be naturally extended • The tradeoff between the cost of additional layers and to the multi-piece information spreading scenario following existing the improvement of information spreading efficiency literature [20]. 3 necessarily the same in each layer) uniformly at random in Remark 1. Given the information spreading time P −γ−1 all layers simultaneously. After each gossip step, the newly Tspr(G, γ), γ > 0, (1 − Tspr (G,γ)(u)) ≤ O(n ) informed nodes (if exist) will broadcast the information to holds for any source node s ∈ V , and any other node all their duplicates. u ∈ V . The equivalence between this new definition of information spreading time and the original one can be 2.2 Information Spreading Time shown by the following inequalities: The metric commonly used to measure the efficiency of gossip based information spreading is the information P r(not all nodes are informed) [ spreading time. Denote St as the informed node set at round = P r( node u is not informed) t S = {s} s ∈ V , with 0 , for some arbitrary . The information u∈V spreading time in a network G of size n, Tspr(G, γ), for X ≤ P r(node u is not informed) some γ > 0, is defined as the stopping time by which all nodes are informed with probability 1 − O(n−γ ) [19], i.e., u∈V −γ X P −γ−1 −γ Tspr(G, γ) = sup inf{t : P r(St 6= V |S0 = {s}) ≤ O(n )}. = (1 − Tspr (γ)(u)) ≤ n × O(n ) = O(n ), s∈V u∈V (3) 3 MAIN RESULTS the first inequality is by union bound. Therefore, the probability that all nodes are informed is Analyzing the information spreading process is difficult −γ even in a single network due to the heterogeneous network at least 1 − O(n ) after time Tspr(G, γ), i.e., Tspr(G, γ) is the stopping time required for all nodes to be informed topology and random gossip processes. To the best of our −γ knowledge, the tightest upper bound for the information with probability 1 − O(n ). spreading time in a general single network in literature is Then, the following definition is needed for the follow- −1 O(Φ · log n) [19, 21], where Φ is the corresponding graph ing analysis: conductance. The multiplex network structure introduces Definition 3. Given a multiplex network G = {Gm , interconnections and interactions among layers, which fur- M (V,Em)}m=1, for each node u ∈ V , dm(u) and Negm(u) ther complicate the analysis. In this study, we slightly relax are defined as the node degree and the neighbor set of the problem and endeavor to find the information spreading u in layer m. The maximum total node degree is then time in a general multiplex network in an idealized setting. M P We are able to obtain a neat result in this setting (Theorem defined as ∆max = max( dm(u)). The total neighbor u∈V m=1 2), which connects the information spreading time to a set Neg(u) of node u is defined as the set of all unique new metric for multiplex networks, multiplex conductance nodes connected to u in any layer, i.e., v ∈ Neg(u) if (Definition 5). In this section, the informed probability M S model is first introduced to facilitate the understanding (u, v) ∈ Em. If v ∈ Neg(u), the link (u, v)’s existing m=1 of information spreading in multiplex networks. Then the 2 layer set is defined as L(u,v) = {α;(u, v) ∈ Eα}, and the corresponding information spreading time will be analyzed. corresponding (u, v) link at layer α is denoted as (u, v)α. 3.1 Informed Probability Model For an information spreading process in a graph G = (E,V ) of size n with source s ∈ V , given the informed In this part, the informed probability model is presented probability set [P (u)] before round t, after gossiping at to offer some preliminary insights of information spreading t,s u∈V round t the informed probability set [P (u)] is given in a multiplex network. In order to facilitate our analysis, t+1,s u∈V by the information spreading time is reformulated. First, the informed probability is defined as: Pt+1,s(u) =(1 − (1 − Pt,pull(u))(1 − Pt,push(u))) P N (4) Definition 1. The informed probability t,s(u), t ∈ is × (1 − Pt,s(u)) + Pt,s(u), ∀u ∈ V, defined as the probability that node u becomes informed P by round t−1 (right before round t) given that the source where t,pull(u) is the probability that node u pulls the node is s. information from its neighbor nodes in round t, and Pt,push(u) is the probability that node u gets the information Based on the above definition, the information spreading by neighbors’ push operation in round t. In the single time is redefined: network, node u successfully pulls the information from one Definition 2. Given the informed probability sets of its neighbors when it contacts a node already informed, P N [ t,s(u)]u∈V , t ∈ defined for information spreading therefore the pull probability is given by in the network G = (V,E) of size n starting at node X 1 s, the corresponding information spreading time can be P (u) = P (v), (5) t,s,pull d(u) t,s defined as v∈Neg(u) −γ−1 Tspr(G, γ, s) = inf{t : max{(1−Pt,s(u))} ≤ O(n )}, u∈V where d(u) is the degree of u. (1) Also, node u successfully gets pushed the information and the information spreading time of network G can be by its neighbors when any of its neighbors already informed alternatively defined as contacts u, so the push probability is given by

Tspr(G, γ) = sup Tspr(G, γ, s). (2) 2 (u, v) s∈V Link may exist in several layers simultaneously. 4 By Eq. (8), the push probabilities of node u in network (1) (2) Y 1 G and G are Pt,s,push(u) = 1 − 1 − Pt,s(v) . (6) d(v) P(1) (u) v∈Neg(u) t,s,push ! 1 For a multiplex network G with M layers, node u Y Y P(1) = 1 − 1 − 1 − 1 − t,s (v) , successfully pulls the information when in any layers dl(v) v∈Neg(1)(u) l∈L(1) it successfully pulls the information. Therefore, the pull (u,v) u G P(2) probability for an uninformed node in network is t,s,push(u) ! M 1 Y X 1 Y Y P(2) Pt,pull(u) = 1 − 1 − Pt,s(v) , (7) = 1 − 1 − 1 − 1 − t,s (v) . d (u) dl(v) m=1 m v∈Neg(2)(u) (2) v∈Negm(u) l∈L(u,v) For the push operation, in each round, node u gets (11) P(1) P(2) Z pushed the information if any of its possible neighbor nodes Next, we need to show t,s (u) ≤ t,s (u), ∀t ∈ , u, s ∈ already informed contacts it. For any v ∈ Neg(u), the V . By induction, first given the same starting node s, probability that v contacts node u is 1 − Q (1 − 1 ). P(1) P(2) P(1) P(2) dl(v) 1,s(u) = 1,s(u), ∀u ∈ V . Then if t,s (u) ≤ t,s (u), l∈L(u,v) P(1) P(2) Therefore, the push probability for an uninformed node u from Eq. (7), t,pull(u) ≤ t,pull(u) since M2 > M1. in network G is Since G(2) is built upon G(1) by adding additional layers !! ! (2) (2) (1) (2) (1) GM +1, ..., GM , then Neg (u) ⊂ Neg (u) and L ⊂ P Y Y 1 P 1 2 (u,v) t,push(u) = 1− 1− 1− 1− t,s(v) . (2) P(1) dl(v) L(u,v), ∀u, v ∈ V . Therefore, from Eq. (8), t,push(u) ≤ v∈Neg(u) l∈L(u,v) P(2) (8) t,push(u). Finally, P(1) (u) Remark 2. Due to the gossip nature, at each layer, an t+1,s P(1) P(1) P(1) uninformed node only attempts to pull the information = (1 − (1 − t,s,pull(u))(1 − t,s,push(u)))(1 − t,s (u)) from one neighbor, but may get pushed the information (1) + P (u) from multiple neighbors. This accounts for the different t,s P(1) P(1) P(2) expressions in Eq. (5) and Eq. (6) when considering the ≤ (1 − (1 − t,s,pull(u))(1 − t,s,push(u)))(1 − t,s (u)) overlapping edges across the layers. P(2) + t,s (u) The informed probability model facilitates the demon- (2) (2) (2) ≤ (1 − (1 − P (u))(1 − P (u)))(1 − P (u)) stration of the beneficial effect of the multiplex network t,s,pull t,s,push t,s P(2) structure qualitatively, as shown in Theorem 1 below. + t,s (u) (1) Theorem 1. Consider a multiplex network G = P(2) = t+1,s(u), ∀u ∈ V. {G(1), ..., G(1) } G(2) 1 M1 , and another multiplex network (12) (1) P(1) P(2) Z built upon G by adding additional M2 − M1 layers, Therefore, t,s (u) ≤ t,s (u), ∀t ∈ , u, s ∈ V . Then the G(2) = {G(1), ..., G(1) ,G(2) , ..., G(2) },M > M s i.e., 1 M1 M1+1 M2 2 1. proof is finished up by contradiction: given source node , (1) (2) (1) Let Tspr(G , γ) and Tspr(G , γ) be the information assume the information spreading times for networks G (1) (2) (2) spreading time for G and G , respectively, then and G are denoted by T1 and T2, respectively. Then by (1) (2) (1) −γ−1 Tspr(G , γ) ≥ Tspr(G , γ). definition max{(1 − P (u))} ≤ O(n ) and max{(1 − u T1,s u P(2) −γ−1 Proof: The key point is to look into the informed T ,s(u))} ≤ O(n ). Also, P 2 probability t,s(u) for each node u. From Eq. (7), the pull (2) (1) (1) max{(1 − P (u))} ≤ max{(1 − P (u))} ≤ O(n−γ−1). probability of node u in network G is u T1,s u T1,s (13) M1 1 P(1) Y X P(1) If T2 ≥ T1, then the above conclusion max{(1 − t,s,pull(u) = 1 − 1 − t,s (v) . u dm(u) (2) −β−1 m=1 (1) P (u))} ≤ O(n ) T = v∈Negm (u) T1,s contradicts the definition 2 (9) P(2) −β−1 inf{t : max{(1 − t,s (u))} ≤ O(n )}. G(2) u Similarly for multiplex network , the pull probability Therefore, for any given source node s ∈ V , of node u is (2) (1) Tspr(G , γ, s) ≤ Tspr(G , γ, s). Finally, by Def. 2, M2 (2) (1) (2) Y X 1 (2) Tspr(G , γ) ≤ Tspr(G , γ). P (u) = 1 − 1 − P (v) t,s,pull d (u) t,s m=1 (2) m v∈Negm (u) 3.2 Idealized Information Spreading Time

M1 Y X 1 (2) Compared to the qualitative analysis above, the quantitative = 1 − 1 − P (v) d (u) t,s analysis for information spreading in the multiplex network m=1 (1) m v∈Negm (u) is much more complicated, which is mainly caused by two M2 1 factors: overlapping edges among layers and heterogeneous Y X P(2) × 1 − t,s (v) . contacting probabilities at different layers. They render the dm(u) m=M1+1 (2) v∈Negm (u) exact estimation of information spreading time in the mul- (10) tiplex network intractable. Therefore, an idealized setting 5 M M P P is considered in the following so that the corresponding where volT (S) = volm(S), |E|T = |Em|, and information spreading time can be analyzed, which serves m=1 m=1 M as a good lower bound for the information spreading time P |cutT (S, V − S)| = |cutm(S, V − S)|. volm(S) is the of the original multiplex network. First, to handle the m=1 overlapping edges among layers, an aggregated multigraph degree sum of all nodes in the node set S at layer m representation for a multiplex network is constructed as (volume), and |cutm(S, V − S)| is the number of edges follows. between node set S and V − S at layer m.

Definition 4. Given a multiplex network G = {Gm , Remark 4. Multiplex conductance coincides with the orig- M inal graph conductance [19] when M = 1. Note that (V,Em)}m=1, the corresponding aggregated multigraph G˜ = (V,˜ E˜) is defined such that V˜ = V and E˜ = the factor of M is deliberately introduced to reflect M , the multiple information spreading processes in the ]m=1Em {(u, v)α; u ∈ V, v ∈ Neg(u), α ∈ L(u,v)}, where ] stands for the non-unique set union, i.e., the multiplex network setting. same links at different layers are all kept. Theorem 2. For an M-layer multiplex network with n nodes, the information spreading time in the idealized setting is −1 1 at most 200(γ + 2)Φmp · (log n + 2 log M) rounds with −γ probability 1 − O(n ), for some γ > 0, where Φmp is the multiplex conductance of this multiplex network. First, the following sequence of random variables solely related to the pull operation is introduced to facilitate our analysis. Let L1,L2, ... be a sequence of random variables with Li (i ≥ 1) defined as follows. We distinguish two cases:

• If volT (Si−1) ≤ |E|T , then by Definition 5, 1 |cutT (Si−1,Ui−1)| ≥ M dΦmpvolT (Si−1)e ≥ 1 M dΦmpvolT (S0)e, where Ui−1 = V − Si−1 is the set of uninformed nodes at round i − 1. Let R = dΦmpvolT (S0)e and Ei be an arbitrary subset M 1 of ]m=1cutm(Si−1,Ui−1) consisting of M R edges. Fig. 1: From Multiplex Network to Aggregated Multigraph Set Ei is (arbitrarily) fixed at the beginning of round i – before the round is executed. Define the minimum volume of a node u as M · min{dm(u)}. For each Remark 3. As shown in Fig. 1, an aggregated multigraph m is obtained from the corresponding multiplex network node u ∈ Ui−1, let δi,u be the 0/1 random variable through a condensation process where the node set with δi,u = 1 if and only if in round i node u pulls remains the same and all links in all layers from the the information through some edge in Ei. Then, original multiplex network are kept. As a result, multiple L = P (δ M · min{d (u)}) i u∈Ui−1 i,u m . u m links can exist between the same pair of nodes and • If volT (Si−1) > |E|T , then Li = R. v in the aggregated multigraph if they are connected Then, the following lemma is needed to prove Theorem in multiple layers of the original multiplex network 2: (e.g., (u, v) ∈ Eα, (u, v) ∈ Eβ, ...). In the aggregated multigraph G˜, these links are considered different and Lemma 1. P P marked as (u, v)α, (u, v)β, ..., respectively. (a) E[ k≤i Lk] = iR and V ar( k≤i Lk) ≤ iR∆max, where R = dΦmpvolT (S0)e. To get around heterogeneous gossip at different layers, (b) If volT (S0) < ∆max, then P r(volT (Si) ≥ ∆max) ≥ node u’s contacting probability for each link (u, v)α is 1/2, for i ≥ 4∆max/(dΦmpvolT (S0)e). 1 unified as P(u) = P((u, v)α) = (denoted min dm(u) m∈{1,...,M} (c) If ∆max ≤ volT (S0) ≤ |E|T then P r(volT (Si) ≥ as link picking probability). This over-optimistic choice min{2volT (S0), |E|T + 1}) ≥ 1/2, for i ≥ 4/Φmp. simplifies our analysis, while still providing a good lower bound as shown below. (d) If volT (S0) > |E|T then P r(volT (Ui) ≤ volT (U0)/2) ≥ The idealized setting is formed through a uniform gossip 1/2, for i ≥ 6/Φmp. with link picking probability P(u), ∀u ∈ V , on the aggregated multigraph G˜ constructed above. The information Proof: The proof of Lemma 1 is provided in the spreading time in this idealized setting for an arbitrary appendix, available in the online supplemental material. multiplex network is quantified below. Remark 5. The random variables Li are used to approximate the increment of volT (Si) over the time. With the Definition 5. The multiplex conductance Φmp of a multiplex , M expectation and variance of the sum of random variables network {Gm (V,Em)}m=1 is defined as Li in Lemma 1(a), different increasing speeds of volT (Si) are shown in Lemma 1(b), (c), and (d), respectively, for |cutT (S, V − S)| three different stages ((d) actually shows the decreasing Φmp = min M , (14) speed of its complement volT (Ui)). S⊂V,volT (S)≤|E|T volT (S) 6

Then, we can prove Theorem 2. distributes the information from vmax to any other nodes in 50(β + 2)Φ−1 · (1 + ∆max )(log n + 1 log M) = 100(β + Proof for Theorem 2: When only the pull operation is m ∆max 2 −1 1 considered, it can be seen from Lemma 1 that the total 2)Φm · (log n + 2 log M) rounds w.h.p. Then by Lemma volume of the informed node set volT (Si) increases in 13 from [19] which shows the symmetry between the pull different ways in the three different stages. The informa- and push operation, the push operation can also spread to tion spreading analysis is then divided into three stages. vmax the information started at an arbitrary source node s in vol (S ) < −1 1 In the first stage, by Lemma 1(b), if given T 0 100(β +2)Φm ·(log n+ 2 log M) rounds with the same high ∆max, after at most d4∆max/(dΦmvolT (S0)e)e ≤ probability. Therefore, the push-pull operation can spread −1 5∆max/(dΦmvolT (S0)e) rounds, the total volume of the the information started at s to all nodes in 200(β + 2)Φm · ∆ 1 informed node set becomes at least max with proba- (log n + 2 log M) rounds w.h.p. bility at least 1/2. Now, divide the information spread- −1 Remark 6. By Theorem 2, we have shown that Θ(Φmp ·log n) ing process in this stage into phases each comprised is a good estimate for information spreading time in 5∆ /(dΦ vol (S )e) of max m T 0 rounds. A phase is consid- a general multiplex network in the idealized setting. It ered successful if the total volume of the informed becomes a true lower bound for the actual information ∆ node set at the end of that phase is at least max. spreading time when the layer number of multiplex d2β ln ne Therefore, in the first phases, the probabil- networks is sufficiently large, as shown below. ity that none of them is successful is at most (1 − 1/2)d2β ln ne ≤ e−dβ ln ne = O(n−β). Therefore, with Theorem 3. Given an M-layer multiplex network G = M at most d2β ln ne · 5∆max/(dΦmvolT (S0)e) ≤ 3β ln n · {Gm}m=1 with n nodes, there always exists a constant −1 5∆max/(dΦmvolT (S0)e) rounds, the total volume of the cn > 0 such that, when M > cn, Θ(Φmp ·log n) is a lower informed node set has volT (St) ≥ ∆max with probability bound of the actual information spreading time. at least 1 − O(n−β). Proof: First, note that the calculated information In the second stage, by Lemma 1(c), if ∆max ≤ −1 spreading time Θ(Φmp · log n) is a decreasing function of volT (S0) ≤ |E|T , then with probability at least 1/2, |cutT (S,V −S)| 1 M ( = Ω( 2 ) = Θ(1) when n is fixed), and it takes at most d4/Φme rounds until the total vol- volT (S) n ume of the informed node set is increased to at least it will go to 0 as M goes to the infinity. Further note min{2volT (S0), |E|T + 1}. Similarly, divide the information that the information spreading time of any scheme for any spreading process in this stage into phases of d4/Φme underlying network topology is lower bounded by some rounds each. A phase is successful if the total volume of constant. Thus, there must exists a constant cn > 0 such −1 the informed node set at the end of the phase is at least that when M > cn, Θ(Φmp · log n) is a lower bound of the min{2volT (Si), |E|T + 1}, where Si is the set of informed actual information spreading time. nodes at the beginning of that phase. Then, for any k, the While Theorem 2 is interesting in theory, in practice, M probability that the k-th phase is successful is at least 1/2, is often a modest number. In this case, if it can be further regardless of the outcome of the previous k − 1 phases. Let shown that the information spreading capability of each B(k, 1/2) denote the binomial random variable for k trials layer is accurately measured (in the order sense) by the −1 each with success probability of 1/2. Then, by the Chernoff corresponding conductance, Θ(Φmp · log n) can still be a bound, the probability that fewer than γ = log |E|T of the good lower bound for the actual information spreading time first k = (2β + 4)γ phases are successful is at most of a multiplex network. Two special cases of special interest are further discussed below. −2(γ−k/2)2/k −βγ −β P r(B(k, 1/2) ≤ γ) ≤ e ≤ e = O(n ), Remark 7. If the single-layer bound is tight for information (15) spreading time in every layer of the given multiplex since |E|T ≥ n − 1. And since at most γ successful phases −1 network (i.e., Tspr,i = Θ(Φi · log n), where Φi is the are required until the total volume of the informed node set conductance of the ith layer), then our bound is also |E| 1 − O(n−β) exceeds T , it follows that with probability tight in the corresponding aggregated multigraph G˜ (i.e., the number of rounds required is at most kd4/Φme ≤ k · −1 T˜spr = Θ(Φ · log n)). 5/Φ ≤ (2β + 4)(2 log n + log M)(5/Φ ) as |E| ≤ M · n2. mp m m T This is intuitively correct by the construction of the vol (S ) > |E| Finally, by Lemma 1(d), if T i T then, with aggregated multigraph and the derivation of Theorem 1/2 d6/Φ e probability at least , it takes at most m rounds until 2. In single networks, the above bound is known the total volume of the uninformed node set is halved. By tight for many graph models of interest. Prominent similar reasoning as before, we can show that once the total examples include complete graphs, random geometric |E| (2β + volume of informed nodes has exceeded T , then graphs (RGGs), and expander graphs (which include 4)(2 log n+log M)(7/Φ ) m rounds suffice to inform all nodes Preferential Attachment (PA) graphs as a special case); with probability 1 − O(n−β). their information spreading times are TC = Θ(log n), Combining all the above three cases and applying log n TRGG = Θ( r ), and TEX = Θ(log n), respectively, the union bound, we obtain that, with probability 1 − 1 with the corresponding conductance given by ΦC = , O(n−β), all nodes get informed within 50(β + 2)(log n + 2 ΦRGG = Θ(r) (r is the radius of RGGs), and ΦEX = 1 log M)(Φ−1 + ∆ /(dΦ vol (S )e) rounds given any 2 m max m T 0 Θ(1). initial informed node set S0 when only the pull operation is considered. Remark 8. If the single-layer bound is essentially tight for Then let vmax be the node of maximum total degree information spreading time in every layer of the given −1 ∆max (see Definition 3). From above, the pull operation multiplex network (i.e., Φ = Θ(D(G)), where D(G) is 7 the diameter of the graph G, and Φ−1 = ω(log n)), then improvement for information spreading in a multiplex our bound is also essentially tight in the corresponding network formed with similar-topology layers would be aggregated multigraph G˜. on the order of M. Our first example below dispels this −1 The information spreading time Tspr = O(Φ · log n) misconception through an innovative design exploiting the in the single network is determined by two factors: the second factor above. inverse of graph conductance Φ−1 and the logarithm of network size log n. It is also known that a spreading 3.3.1 Proposed Ring-Ring Multiplex Network Ω(D(G)) algorithm in any graph needs at least rounds In this example, a novel ring-ring multiplex network struc- Φ−1 = Θ(D(G)) to finish. If , it indicates that the inverse ture is proposed to demonstrate the impact of increased of graph conductance is on the same order as the graph contacting opportunities in a multiplex network, for which log n √ diameter. The extra in the original bound is due to an improvement as large as Θ( n) can be achieved. In this the distributed nature of the gossip algorithm, therefore proposed multiplex network structure (as shown in Fig. 2), it is essentially tight or gossip is essentially effective in nodes are identified with the integers 1, 2, ..., n. In the first such graph structures. Ring graph is such an example, layer, each node with ID i connects with two nodes with for which the information spreading bound is essentially 2 IDs ((i + 1) mod n) and ((i − 1) mod n), which forms tight, since its conductance Φring = n and diameter n a normal ring structure. Next, assuming without loss of Dring = . 2 generality that n + 1 is√ not a prime number, find out its Many types of graphs as pointed out belong to the factor r that is closet to n. Then in the second layer, node above two categories, except for star-like graphs, for which with ID i is connected with two nodes with IDs ((i + r) the conductance Φstar = 1 and the information spreading mod n) and ((i − r) mod n) instead. Consider the right time Tstar = 1. Therefore, we expect that a multiplex connection of each node, then in the second layer starts from n+1 network comprised of component layers constructed (or node with ID 1, after r such connections, it will reach the −1 (1 + (n + 1)) mod n 2 well modeled) by the above graphs will admit Θ(Φmp·log n) node with ID which is . The same as a good lower bound for information spreading time3. process repeats for n times, the right connections will reach In this case, the improvement in information spreading back to node 1, therefore the second layer is guaranteed with efficiency from a single network to a multiplex network can a ring structure. Φmp be orderly determined by PI = Φ .

3.3 Multiplex Conductance Evaluation and Perfor- mance Improvement Our main result above, as a generalization and advance of the current art of the single network study, nicely encapsulates the gossip-based information spreading performance in a multiplex network into a metric, its multiplex conductance. In this part, the multiplex conductance is evaluated for some interesting multiplex networks to facilitate understanding. As can be seen from Definition 5, the multiplex conductance is calculated through jointly minimizing the overall Fig. 2: Proposed Ring-Ring Structure r = 4 cut-volume ratio of the multiplex network. Evaluation of the single network conductance is in general a hard problem Then the multiplex conductance of this network struc- already, and existing results are mainly obtained for graphs ture is given by the following corollary: with certain nice properties and usually in the order form Corollary 1. The multiplex conductance of the above [20, 22]. The existence of multiple layers and interactions proposed ring-ring multiplex network is Φmp = 2 min{r+1,(n+1)/r+1} among layers further complicates the evaluation. Therefore, . as a first work in this area, we will focus on evaluating some n multiplex networks with special structures like ring graphs, Proof: The proof of Corollary 1 is provided in the which allow us to obtain some inspiring results. The study appendix, available in the online supplemental material. of the multiplex conductance for multiplex networks with In the proposed ring-ring structure, nodes in two layers other types of network topologies will be investigated in maintain completely different connections, which is close to our future work. Based on the definition of multiplex con- the idealized setting in the previous discussion. Therefore, ductance, two key factors contribute to higher information as shown in Fig. 3, the predicted information spreading effi- spreading efficiency in multiplex networks as compared to Φmp ciency improvement PI = Φ = min{r + 1, (n + 1)/r + 1} the single network: 1) availability of multiple channels M, is quite tight in this scenario. |cut (S,V −S)| and 2) larger contacting opportunities T . There volT (S) Remark 9. The second layer of the proposed ring-ring may be a common misconception that the best achievable structure dramatically increases the size of cut (e.g., the size of the half-half cut is 2 in the original network but 3Note that different layers may have different structures (such as one PA layer coupled with one RGG layer) so long as the single-layer it becomes 2r + 2 in the proposed ring-ring structure) bound is tight in each. through the rewiring of nodes without increasing the 8

350 7000 Simulation Ring-PA (Simulation) Φm 300 Φ 6000 Ring-Complete (Simulation) (n) 250 5000

200 4000

150 3000

100 2000 Efficiency Improvement Efficiency Improvement

50 1000

0 1 2 3 4 5 0 10 10 10 10 10 0 2 4 6 8 10 Network Size Network Size 104 Fig. 3: Proposed Ring-Ring Fig. 4: Ring-Complete and Ring-PA

set volume. This smart design leads to an informa- α tion spreading efficiency improvement on the√ order of where is the edge expansion of the PA graph. The min{r + 1, (n + 1)/r + 1}, which is close to n in most last two inequalities are by the expansion properties of cases. PA graphs [22]. Again the performance improvement is Θ(n). Both results are verified in Fig. 4. 3.3.2 Different-Structure Multiplex Network This example reveals increased contacting opportunities in 3.3.3 Identical-Graph Multiplex Network a multiplex network from a new perspective. Note that multiplex networks don’t need to be constrained by con- Finally, the impact of layer number M is explored, construction of similar-topology layers. Intuitively, a marriage sidering a special type of multiplex networks formed with of different structures can lead to a dramatic improvement M layers of identical topology. When M identical graphs in information spreading for the disadvantaged layer, as having conductance Φ form a multiplex network, the overall shown by the following two cases. multiplex conductance can be shown as • Ring-Complete Coupling: If a ring graph is coupled with another complete graph to form a ring-complete multiplex network, the corresponding multiplex con- M · |cut(S, V − S)| Φmp = min M · ductance may be computed as S∈V,volT (S)≤|E|T M · vol(S) (18) 2 + |S|(n − |S|) = MΦ. Φmp = min 2 = Θ(1). (16) |S|≤n/2 2|S| + |S|(n − 1) Therefore, an improvement of order M is expected for Φmp information spreading efficiency in this setting, which is Then the performance improvement is PI = Θ( Φ ) = Θ(n). generally an over-estimate, as the effect of link collision at • Ring-PA Coupling: Consider a PA graph with the pa- different layers is ignored. This effect is most severe for the rameter d, i.e., a new node connects to d existing nodes identical-layer structure, and can be partially corrected by with probabilities proportional to their degrees, then considering the average meaningful contact for each node. the corresponding multiplex conductance of a ring-PA For a node i with degree d in each layer, the average multiplex network is given by i number of meaningful contacts in each time slot after 1 M |cutr(S, V − S)| + |cutPA(S, V − S)| eliminating duplicate contacts is di(1−(1− d ) ). Therefore, Φmp = min 2 i S∈V, volr(S) + volPA(S) the information spreading efficiency improvement for the vol (S)≤|E| T T whole network is upper bounded by ∆(1−(1− 1 )M ), where 2 + |cut (S, V − S)| ∆ ≥ min 2 PA ∆ is the largest node degree across all layers in the network. S∈V, 2|S| + volPA(S) volT (S)≤|E|T By dividing the single layer information spreading time by 1 M |cut (S, V − S)| M and ∆(1 − (1 − ) ), respectively, the idealized and ≈ min 2 PA ∆ S∈V, 2|S| + volPA(S) corrected information spreading time can be calculated as volT (S)≤|E|T shown by the bottom three grouped curves (with diamond |cut (S, V − S)| ≥ min 2 PA markers) and middle three grouped curves (with square S∈V, 2|S| + d|S| + |cutPA(S, V − S)| markers) in Fig. 5. When ∆ is small (like in the ring graph), volT (S)≤|E|T 2α this simple correction leads to an improvement as shown in ≥ = Θ(1), 2 + d + α Fig. 5. For multiplex networks constructed by independent (17) layers, the over-estimation error is usually not a concern. 9

104 7 1800 1-Layer (Simulation) Ring−PA (Average) 2-Layer (Simulation) 1600 Ring−Ring (Proposed) 6 3-Layer (Simulation) 4-Layer (Simulation) Ring−PA (Minimum) 2-Layer (Corrected) 1400 Ring−Ring (Identical) 3-Layer (Corrected) 5 4-Layer (Corrected) Ring−Complete 2-Layer (Idealized) 1200 3-Layer (Idealized) 4 4-Layer (Idealized) 1000

3 800

600

2 Reward−Cost Ratio Information Spreading Time 400 1 200

0 0 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 Network Size 104 4 Network Size x 10

Fig. 5: Information Spreading in Multiplex Networks with Fig. 6: User Aspect Identical Ring Graphs

120 4 LAYER COSTAND INFORMATION SPREADING Ring-Ring (Proposed) EFFICIENCY Θ1(√n) 100 Ring-PA (Minimum) In real life, the adoption of a new layer comes with an Θ2(√n) additional cost. In this section, the tradeoff between the Ring-Ring (Identical) improvement of information spreading efﬁciency and the 80 Ring-Complete additional layer cost is discussed. In particular, we will exam two types of cost below.4 60 • Network Cost for a User: The cost of additional layers from a user u’s aspect is measured by the total degree 40

PM Reward−Cost Ratio of this node in all layers, i.e., CML(u) = m=1 dm(u). Therefore, the corresponding cost increase for the user u 20 PM m=1 dm(u) is CI,U (u) = d(u) , where d(u) is the node degree

of u in the initial single network. 0 0 2 4 6 8 10 • Network Cost for the Network Designer: Different from the 4 Network Size x 10 user’s aspect, for a network designer, the cost of a new layer is better measured by the number of total edges Fig. 7: Enlargement of Fig. 6 of it. Then the cost increase for the network designer PM m=1 |Em| is CI,N = |E| , where |E| is the number of total edges of the initial single network. ring-complete coupling brings dramatic improvement With the cost increment CI for each multiplex network, in information spreading efficiency, the associated cost and the corresponding information spreading efficiency somewhat offsets its effectiveness. improvement PI , the reward-cost ratio can be defined as 3) From a ring graph to a two-layer ring-preferential- RC = PI . In the following, four related cases are examined attachment (PA) multiplex network: In this case, the CI to shed some light on the tradeoff between performance performance improvement is PI = Θ(n) by Eq. (17). improvement and cost: But the cost increment is now different for the user 1) From a ring graph to a two-identical-ring multiplex and network designer, since the cost increments are different for different nodes due to network irregularity. network: In this case, the information spreading effi- max ciency improvement is 2. The cost increments for both Both the maximum cost increment CI,U and average cost increment Cave are considered for the user. Since the user and the network designer are CI,U = CI,N = I,U the largest node degree in the PA network with n nodes 2. Then the reward-cost ratios are RCU = RCN = 1. √ grows as Θ( n) [23], the maximum cost increment is 2) From a ring graph to a two-layer ring-complete mul- max √ tiplex network: In this case, the performance improve- CI,U = Θ( n). However, the average node degree for Φmp 2d ment is P = Θ( ) = Θ(n) by Eq. (16). The cost the PA network is . Then the average cost increment I Φ Cave = d + 1 increments for both the user and the network designer is I,U . Therefore, the minimum reward-cost n+1 ratio and average reward-cost ratio for the user are are CI,U = CI,N = 2 . Then the reward-cost ratios min √ ave RCU = Θ( n) and RCU = Θ(n), respectively. are RCU = RCN = Θ(1). Therefore, even though the For the network designer, the cost increment is CI,N = 4The choices of cost in this paper mainly serve to facilitate relevant d + 1, and the reward-cost ratio is RCN = Θ(n). discussion. In practice, other meaningful costs may also be considered. Clearly, the ring-PA coupling is much more effective 10

1800 120 Ring−PA Ring-Ring (Proposed) 1600 Ring−Ring (Proposed) Θ(√n) Ring−Ring (Identical) 100 Ring-Ring (Identical) 1400 Ring−Complete Ring-Complete 1200 80

1000 60 800

600 40 Reward−Cost Ratio Reward−Cost Ratio 400 20 200

0 0 0 2 4 6 8 10 0 2 4 6 8 10 4 Network Size 4 Network Size x 10 x 10

Fig. 8: Designer Aspect Fig. 9: Enlargement of Fig. 8

as compared to the ring-complete coupling when the communication networks. Despite of the effectiveness layer cost is explicitly considered. in spreading information, its existence requires a huge 4) From a ring graph to the ring-ring structure proposed amount of management and computation resource. The in 3.3: It has been shown that the efficiency improve- ring network, which has a much smaller number of ment is PI = min{r + 1, (n + 1)/r + 1}. The cost edges, a large diameter, and is very poor-connected, is increments for both the user and the network designer chosen as another extreme but realizable case opposed are CI,U = CI,N = 2, then the reward-cost ratios are to the PA network. And, in reality, it can correspond to RC = RC = min{r+1,(n+1)/r+1} the type of networks that a small group of people want to √U N 2 , which is close to Θ( n) in most scenarios. build to facilitate their own information spreading with a limited budget (small number of edges) and the real Remark 10. Figs. 6 and 7 simulate the actual reward- spatial constraint (poor-connected) on top of the existing cost ratios from the user’s perspective, while Figs. 8 social and communication networks. and 9 present the results from the network designer’s viewpoint. From both the discussion above and simula- tions, it can be seen that, overall speaking, the ring-PA 5 CONCLUSIONAND FUTURE WORK coupling is the most effective multiplex structure as far as information spreading is concerned. Since PA graph is In this work, the gossip based information spreading is a popular model for social platforms, this result partially studied in multiplex networks. By defining the new metric −1 reveals the beneficial impact of utilizing social networks multiplex conductance Φmp, Θ(Φmp · log n) is found to for information distribution. be a good estimate for information spreading in many On the other hand, the preferential attachment design is multiplex networks of interest. The multiplex conductance much more complicated as compared with our proposed is then evaluated for several interesting multiplex networks ring-ring structure. Therefore, as far as the overall net- to help understand information spreading potentials of work design is concerned (instead of utilizing existing multiplex networks. By further taking the additional layer network structures), our proposed structure provides the cost into consideration, the tradeoff between the cost and the insight that a careful planning of the new layer structure information spreading efficiency improvement is discussed can greatly improve the overall efficiency for information from both the user’s and the network designer’s perspective spreading without incurring much additional cost. to facilitate the understanding in this burgeoning area. The goal of this section is to study the benefit of facili- Many interesting problems remain open in this research tating the information spreading in multiplex networks field. We are able to estimate the information spreading through additional layers. We picked three different time only with the idealized setting. In future work, we topologies, ring graph, complete graph, and PA graph, plan to estimate the actual information spreading time for as candidates for each layer of multiplex networks. the gossip based model in multiplex networks. And we The complete graph represents the idealized single also plan to extend this study to more general network network structure which serves as a base line compared settings like multilayer networks. Meanwhile, the expected to any other topologies. The PA graph, which has a spreading time and extinction time has not been studied in large amount of edges, a very small diameter, and is the other information spreading model-SI/SIR like epidemic well-connected, is chosen as an extreme but realizable model, which will be our focus in the next step. Finally, topology for a single network. In reality, it corresponds we plan to look into other interesting networking problems to the social network that greatly speeds up the in- like community detection and influence maximization in formation spreading process on top of the traditional multilayer and multiplex network settings. 11

ACKNOWLEDGEMENT [22] M. Mihail, C. Papadimitriou, and A. Saberi, “On certain connectiv- ity properties of the internet topology,” in Foundations of Computer This work was supported in part by National Science Foun- Science, 2003. Proceedings. 44th Annual IEEE Symposium on. IEEE, dation under Grants ECCS-1307949 and EARS-1444009, and 2003, pp. 28–35. in part by Army Research Ofﬁce under Grant W911NF-17- [23] A.-L. Barabasi,´ “Network science book,” Network Science, 2014. 1-0087.

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