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Condensing Temporal Networks Using Propagation Condensing Temporal Networks using Propagation Bijaya Adhikari∗ Yao Zhang∗ Aditya Bharadwaj∗ B. Aditya Prakash∗ Abstract Modern networks are very large in size and also evolve with time. As their size grows, the complexity of performing net- work analysis grows as well. Getting a smaller representa- tion of a temporal network with similar properties will help in various data mining tasks. In this paper, we study the novel problem of getting a smaller diffusion-equivalent rep- resentation of a set of time-evolving networks. Figure 1: Condensing a Temporal Network We first formulate a well-founded and general temporal- network condensation problem based on the so-called been recently studied, surprisingly getting a smaller rep- system-matrix of the network. We then propose NetCon- resentation of a temporal network has not received much dense, a scalable and effective algorithm which solves this attention (see related work). Since the size of temporal problem using careful transformations in sub-quadratic run- networks are order of magnitude higher than static net- ning time, and linear space complexities. Our extensive ex- works, their succinct representation is important from periments show that we can reduce the size of large real a data compression viewpoint too. In this paper, we temporal networks (from multiple domains such as social, study the problem of `condensing' a temporal network co-authorship and email) significantly without much loss of to get one smaller in size which is nearly `equivalent' information. We also show the wide-applicability of Net- with regards to propagation. Such a condensed net- Condense by leveraging it for several tasks: for example, work can be very helpful in downstream data mining we use it to understand, explore and visualize the original tasks, such as `sense-making', influence maximization, datasets and to also speed-up algorithms for the influence- immunization and so on. Our contributions are: maximization problem on temporal networks. • Problem formulation: Using spectral characteriza- tion of propagation processes, we formulate a novel 1 Introduction and general Temporal Network Condensa- problem. Given a large time-varying network, can we get a tion • Efficient Algorithm: We design careful transforma- smaller, nearly \equivalent" one? Networks are a com- tions and reductions to develop an effective, near- mon abstraction for many different problems in various linear time algorithm which is also domains. Further, propagation-based processes are very NetCondense easily parallelizable. It merges unimportant node useful in modeling multiple situations of interest in real- and time-pairs to quickly shrink the network with- life such as word-of-mouth viral marketing, epidemics out much loss of information. like flu, malware spreading, information diffusion and • Extensive Experiments: Finally, we conduct multi- more. Understanding the propagation process can help ple experiments over large diverse real datasets to in eventually managing and controlling it for our benefit, show correctness, scalability, and utility of our al- like designing effective immunization policies. However, gorithm and condensation in several tasks e.g. we the large size of today's networks, makes it very hard to show speed-ups of 48 x in influence maximization analyze them. It is even more challenging considering over dynamic networks. that such networks evolve over time. Indeed, typical mining algorithms on dynamic networks are very slow. The rest of the paper is organized in the usual way. We One way to handle the scale is to get a summary: omit proofs due to lack of space. the idea is that the (smaller) summary can be analyzed 2 Preliminaries instead of the original larger network. While summa- We give some preliminaries next. Notations used and rization (and related problems) on static networks has their descriptions are summarized in Table 1. Temporal Networks: We focus on the analysis of ∗Department of Computer Science, Virginia Tech. Email: dynamic graphs as a series of individual snapshots. fbijaya, yaozhang, adb, [email protected]. In this paper, we consider directed, weighted graphs Table 1: Summary of symbols and descriptions tence of many nodes/edges which are not important in Symbol Description propagation. Similarly, as changes are typically grad- G Temporal Network ual, most of adjacent time-stamps are not drastically Gcond Condensed Temporal Network different. There may also be time-periods with sparse th Gi, Ai i graph of G and adjacency matrix connectivities which will not contribute much to prop- wi(a; b) Edge-weight between nodes a and b agation. Overall, these observations intuitively imply in time-stamp i that it should be possible to get a smaller `condensed' V ; E Node-set; Edge-set representation of G while preserving its diffusive char- αN Target fraction for nodes αT Target fraction for time-stamps acteristics; which is our task. T # of timestamps in Temporal Network It is natural to condense as a result of only local FG Flattened Network of G `merge' operations on node-pairs and time-pairs of G| XG Average Flattened Network of G such that each application of an operation maintains SG The system matrix of G the propagation property and shrinks G. This will also F ; X The adjacency matrix of F ; X G G G G ensure that successive applications of these operations λS Largest eigenvalue of SG λF; λX Largest eigenvalue of FG ; XG `summarize' G in a multi-step hierarchical fashion. A Matrix (Bold capital letter) More specifically, merging a node-pair fa; bg will u, v Column Vectors (Bold small letter) merge nodes a and b into a new super-node say c, in all Gi in G. Merging a time-pair fi; jg will merge G = (V; E; W ) where V is the set of nodes, E is the graphs Gi and Gj to create a new super-time, k, set of edges and W is the set of associated edge-weights and associated graph Gk. However, allowing merge w(a; b) 2 [0; 1]. A temporal network G is a sequence of T operations on every possible node-pair and time-pair graphs, i.e., G = fG1;G2;:::;GT g, such that the graph results in loss of interpretability of the result. For at time-stamp i is Gi = (V; Ei;Wi). WLOG, we assume example, it is meaningless to merge two nodes who every Gi in G has the same node-set V (as otherwise, if belong to completely different communities or merge T we have Gi with different Vi, just define V = [i=1Vi). times which are five time-stamps apart. Therefore, we Our ideas can, however, be easily generalized to other have to limit the merge operations in a natural and types of dynamic graphs. well-defined way. This also ensures that the resulting Propagation models: We primarily base our dis- summary is useful for downstream applications. We cussion on two fundamental discrete-time propaga- allow a single node-merge only on node pairs fa; bg such tion/diffusion models: the SI [3] and IC models [10]. that fa; bg 2 Ei for at least one Gi, i.e. fa; bg is in the The SI model is a basic epidemiological model where unweighted `union graph' UG(V; Eu = [iEi). Similarly, each node can either be in `Susceptible' or `Infected' we restrict a single time-merge to only adjacent time- state. In a static graph, at each time-step, a node in- stamps. Note that we can still apply multiple successive fected/active with the virus/contagion can infect each merges to merge multiple node-pairs/time-pairs. Our of its `susceptible' (healthy) neighbors independently general problem is: with probability w(a; b). Once the node is infected, it stays infected. SI is a special case of the general ‘flu- Informal Problem 1. Given a temporal network G = like' SIS model, as the `curing rate' (of recovering from fG1;G2;:::;GT g with Gi = (V; Ei;Wi) and target frac- the infected state) δ in SI is 0 while in SIS δ 2 [0; 1). tions αN 2 (0; 1] and αT 2 (0; 1], find a condensed cond 0 0 0 0 In the popular IC (Independent Cascade) model nodes temporal network G = fG1;G2;:::;GT 0 g with Gi = 0 0 0 get exactly one chance to infect their healthy neighbors (V ;Ei;Wi ) by repeatedly applying \local" merge op- with probability w(a; b); it is a special case of the gen- erations on node-pairs and time-pairs such that (a) 0 0 cond eral `mumps-like' SIR model, where nodes in `Removed' jV j = (1 − αN )jV j; (b) T = (1 − αT )T ; and (c) G state do not get re-infected, with δ = 1. approximates G w.r.t. propagation-based properties. We consider generalizations of these models to temporal networks [17], where an infected node can only 3.1 Formulation framework Formalizing Informal infect its susceptible `current' neighbors (as given by G). Problem 1 is challenging as we need to tackle following Note that models on static graphs are special cases of two research questions: (Q1) Characterize and quantify the propagation-based property of a temporal network those on temporal networks (with all Gi 2 G identical). G; (Q2) Define \local" merge operations. 3 Our Problem Formulation In general, Q1 is difficult as the characterization Real temporal networks are usually gigantic in size. should be scalable and concise. For Q2, the merges However, their skewed nature (in terms of various dis- are local operations, and so intuitively they should be tributions like degree, triangles etc.) implies the exis- defined so that any local diffusive changes caused by them is minimum. Using Q1 and Q2, we can formulate worrying about the system matrix. For µ(G; i; j), the Informal Problem 1 as an optimization problem where `locality of change' is Gi, Gj and the new Gk.
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