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: fyhuang20, [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 fGm , (Vm;Em)gm=1 that constitute the layers of this complex system, together with the interlayer unique nodes connected to u in any layer, i.e., v 2 Neg(u) M connections represented by Eαβ, for any two different layers S if (u; v) 2 Em. If v 2 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) = fα;(u; v) 2 Eαg, 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 = f(vα; vβ); v 2 V g 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 = fGm , attempts to grab the information. In this work, we will consider M ~ (V; Em)gm=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 f(u; v)α; u 2 V; v 2 Neg(u); α 2 L(u;v)g, 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) m2f1;:::;Mg 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 fsg, for some arbitrary s 2 V . The information spreading with link picking probability P(u); 8u 2 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 infft : plex network is quantified below. s2V −γ Definition 3: The multiplex conductance Φmp of a multiplex P r(St 6= V jS0 = fsg) ≤ O(n )g. M network fGm , (V; Em)gm=1 is defined as III. MAIN RESULTS jcutT (S; V − S)j 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)≤|EjT volT (S) heterogeneous network topology and random gossip processes. M M P P The multiplex network structure introduces interconnections where volT (S) = volm(S), jEjT = jEmj, 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 jcutT (S; V − S)j = jcutm(S; V − S)j. 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.